tl;dr: Go is a game that is fundamentally about flexibility, reading and positional judgement. From these arise concepts such as exchange, urgency, big moves, thickness etc… Players across the spectrum have various degrees of mastery in these fundamental things.
Having had a two year journey starting at the bottom of the TPK ladder to somewhat of a solid shodan player, here’s what I have come to learn about the rank ranges.
Of course, these are my personal observations only. It is by no means an objective statement.
TPK (25k-20k)
New players who are still trying to understand the bare basics of stones, edges and corner intersections.
Typical characteristics:
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Will try to save every stone
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Will try to run out an unfavorable ladder
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Sometimes do not see ataris or self ataris
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Next to no concept of territory or moyo or anything beyond what is physically present on the board
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Highly devoted to capturing stones and avoid being captured
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No concept of weaknesses, shape or haengma
DDK (19k - 10k)
Players in this range have either played for merely few weeks or months to years or even decades.
Key improvements from TPK:
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Recognized Go is about making points than capturing stones.
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Knows the bare minimum of life and death
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Knows few commonly played shapes to maintain connection or break connection (usually by experience from past games)
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Rough idea of what is territory, what is moyo (especially 13k+)
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Understands basic go theory and few common josekis
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Will sometimes defend a cutting point in their own position
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Knows how to use some basic shortage of liberty aji during endgame.
Bad habits and weaknesses that leads to rank stagnation:
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Weak understanding of shape (no concept of vital points of stones)
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Laziness (playing moves that seem to work from past experience instead of reading at every move to make an informed choice)
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Inflexible thinking (I have stones here so this area must be my territory, if you invade in my area I will kill you. If you reduce me I will hold on to what I have instead of making a trade.)
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No concept of reasonableness (Will invade in gaps larger than two spaces apart), this is a common habit further encouraged by the opponent not knowing how to punish severely.
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Tends to follow where the opponent played last
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Pays no attention to the whole board position, thinks very locally.
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Plays aji keshi and erases almost all good aji in every position.
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No concept of stone efficiency
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Creates extremely heavy groups everywhere, no concept of light play or sacrifice.
I want to say that the longer a player remained DDK the more profound these mistakes are.
Weak SDK (OGS 9k - 4k)
Players who have played for a few months or a year can usually reach this stage if they are critical of their games and keeps an open mind for learning new things.
This is also by far the most volatile level gap. There are players of all styles and habits here because different people would have spent different amount of time at this stage and accumulated certain mindsets.
Improvements from DDK:
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Better reading skills overall, most actually reads a few moves ahead somewhat accurately.
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Better understanding of common tesuji and can utilize them to get out of certain situations.
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Understanding basic direction of play and knows how to build a position for themselves.
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Understands which moves are big and urgent to a degree and can apply these occasionally.
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Understands some common shapes for corner enclosure, extension, invasion, reduction and just shapes overall in middle game. Also has an awareness for large points and urgent points.
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Understands, to a certain degree, strength and weakness of groups and will have an awareness to defend a weak group. Knows common techniques to attack in the middle game.
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Understands the concept of exchange and is sometimes willing to exchange one dead group for another. Will not always passively defend own territory.
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Has a general idea of how to use influence.
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Is willing to sacrifice a stone or two, or maybe three for the greater good…
Pitfalls:
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Because of their over confidence in escape tesuji and their ability to escape or make life, SDK s at this range is the most likely to invade aggressively everywhere. Opponents of equal strength usually finds these invasions hard to catch and difficult to punish. Playing like this is one major reason a SDK find their rank stagnate.
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Some SDKs are heavily focused on destroying the opponents position instead of building their own throughout the entire game. Some may find this style annoying, but dealing with it is a test of your go fundamentals. This is the second major factor of rank stagnation at this stage.
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Sometimes will choose an inadequate shape to defend or attack, leading to future problems as the game develops.
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Sometimes overlooks a weakness in their own position when attacking or defending.
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Highly unreasonable (some SDK players), wants everything in the board and wants you to have nothing. This almost never works out well.
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Overly aggressive to the point of overplay, but being able to punish takes a greater deal of effort, which is why they can sometimes get away with it especially against weaker players.
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Often will misjudge the result of a local exchange and make unfavorable exchanges.
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Still have little understanding of preserving aji and will play aji keshi and crude moves.
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Can still be inflexible with decision making. Will often stubbornly hold on to what they have instead of venturing out to attack. Reluctance to sacrifice stones is one major trend.
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Some players overplay to solely exploit mistakes of the opponent (whether due to time pressure or aji) instead of playing something that actually works.
Strong SDK (3k - 1k)
By now these players know most of the fundamentals.
Improvements:
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All of the aforementioned improvements done better.
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More flexibility in their thinking, willingness to exchange and trade.
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Occasionally mindful of sente and gote. Will often try to defend in sente.
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Better judgment of board position and a growing awareness of thickness and thinness of groups.
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Usually have a fine instinct for common tesuji points and haengma vital points.
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Basic awareness of aji, but will still play many crude or aji keshi moves.
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Familiar with lighter play when invading deep in enemy territory. Knows to create aji with attachments etc. Has a degree of flexibility in their moves when the situation demands it.
Pitfalls:
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Still unreasonably aggressive sometimes. Often will overcommit to an attack that may or may not work… Will often overplay to kill. Usually my games against opponents of this level ends up with a dead group because of an overaggressive attack.
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Will still invade occasionally in questionable places. Though this will occur less than those in the last group.
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Questionable joseki choice or direction of play. Often still thinks locally.
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Sometimes will make questionable shapes in attack and defense.
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Still poor evaluation of local exchanges. Will play exchanges that leads to a degree of local loss.
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Sometimes makes questionable haengma that can be exploited in some way, leading to a totally collapsing position.
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Sometimes plays slow moves
Low dans (1d - 3d)
The one biggest misconception I had for a long time as a DDK player is that dan level players can make magic happen out of seemingly nowhere. At the end of the day, they are still players who cannot bend the rules of the game. Instead of getting caught in a tricky spot and think a dan level player can find a magical tesuji to reverse it all, what makes dan level players generally stronger is that they can anticipate the danger and avoid it before it happens. This is a result of game experience and honed intuition.
If anything I just want to emphasise 1 thing low dans do more consistently compared to SDKs:
- In some situations, they will invest one move to defend themselves in gote and recognise that is the largest move on the board
Everyone loves to attack and enjoy the thrill of watching the opponent squirm. But sometimes the secret to strength is the most mundane 
That being said, low dans make low dan mistakes. Much of the comparison used herein is in comparison against the high dan standard.
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Unwillingness to sacrifice former attacking stones. Does not recognise consistently that vital attacking stones that were useful a few moves ago can be worthless and not need saving a few moves later. This results in the creation of a heavy group – a burden – for the rest of the game.
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Though decent with local josekis, will still make questionable joseki choices.
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Often lacks consideration of full-board position and play sequences that are equal locally but doesn’t work with rest of the stones on the board.
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Unstable grasp of using aji.
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Unsensitive to shape of stones and the aji in said shapes.
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Will still make unfavourable exchanges that give opponent good shape, or weaken (or even erase) the aji in opponent’s position.
High dans (4d - 6d+)
I happen to be in a club full of 4d+ players (and I’m the weakest player there). From all the games I played against them, I can safely say the following:
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They all have a solid grasp of Go fundamentals
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They are good at handling weak groups to an extent where you cannot capture them (save for a blunder which still happens rarely).
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They know almost all the common and special tesujis during local fights
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They are familiar with vital points of attack and defense in the middle game
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They have better endgame compared to low dans, but still questionable sequences by professional standards. Endgame is a common weakness amongst amateur players.
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Because of subjectivity in positional judgement, one player might end up losing out in a position when they think they’re still ahead.
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They will make an effort to count territory and analyse their position regularly and develop their strategy.
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Much more flexible than previous group, but still not flexible enough by professional standards 
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Understands thinness and thickness of a position
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Will have the right ‘direction’ and ‘instinct’ to attack, but the exact stone location can often be better by professional standards.
Games between high dans comes down to which players are making more efficient exchanges compared to another. If one player can take advantage of the others’ shape and gain 3 points in sente, it would be a decent lead. Since high dans are usually players who played for a few years, each of them understands the game in a very different way. High dans usually argue over whether a given position was good for white or black based on their biased views on the game. There’s plenty of that at the club :^)
As for even stronger players…
A Chinese 4p once said: “A player many stones stronger than you will make you feel disgusted at your position. They will demand you to settle your stones in a way that is highly inefficient, but not settling it will be even worse. It’s a very uncomfortable feeling, but that’s what professionals do. They don’t aim to capture you, but they want their stones to work better than yours.”
I am not as qualified to comment on players around this range since I haven’t been there yet. However, from many lessons with my 6d+ go teacher, whose strength is on par with a weak professional player, I can infer the following:
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They are able to read deep and read accurately in any given position
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They are highly sensitive to the shape of stones and the aji in said shapes.
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They are able to play moves that threaten to utilise aji in their zombie stones.
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They are flexible with their position and is usually willing to sacrifice stones for a larger gain. In particular, they value thickness more than lower dans because they can make better use of them.
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They can judge a position quite accurately and recognise which player has gained or lost from an exchange locally. Though this is still highly subjective, which is the cause for strength disparity among stronger players.
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Wins or losses at this stage is usually determined by who has better reading and strategy.
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Their life and death is very robust; given 2 minutes they can read most common local positions if they don’t know them by heart already.
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Very cunning abuse of byo-yomi time controls to maximize thinking time 
Some final remarks
Below are some of my opinions on the nature of Go. Personally I see some truth to these opinions, though I would encourage people to discuss it and share their thoughts, perhaps in another separate thread.
What makes the difference between players?
- Judgement
In my honest opinion, go is fundamentally a game about reading and judgement. Kobayashi Koichi 9p loves territory, Takemiya Masaki 9p favors influence. Rumour has it that an amateur asked both players: “What’s the weakness of the star point?”. To which Kobayashi responded: “The 3,3 invasion”, and Takemiya said: “The cover at 5,5”. Before AIs arrived in the go scene, it’s hard to debate who is right and who is wrong at the top level of play. But given a position, how you understand it plays a vital role in how you plan to play your subsequent moves. Whether you play risky because you think you’re behind, or you play solid because you think you’re ahead. At the top level, every point matters, which is why judgement is extremely important.
Of course, as amateurs, the first step is to be able to ‘judge’ a position. Then it’s a matter of judging accurately, and judging fast. Judge a position globally to see who’s ahead territorially, who has thickness, where are the thin areas in opponent’s position. Next it’s the ability to evaluate the relative gains and losses in between exchanges. Did white gain more than black after a local non-joseki exchange? Without these concepts, one would be playing moves they think were necessary when in fact they were losing lots of points. In the end they might not even know why they lost!
- Reading
Judgement has its foundations on reading. Without accurate and deep reading, you cannot judge accurately whether a variation is good or bad for you. And Go players, amateur and professionals alike, dedicate their life to hone these two fundamental skills. Life and death is a must for all who seeks to improve.
- Intuition and creativity
Everyone, given enough time and study, will understand common shapes for attack, defense, sabaki and the like. But those with creative intuition can see unconventional moves that most people won’t consider. Often times these creative moves can throw the opponent off-guard, or be a straight-up beautiful tesuji that the opponent did not consider when playing his move. Keep an open eye for moves on the board beyond the conventional shapes. Not all empty triangles are bad moves!
- Flexibility
All the above points lead to the final one. You don’t have to connect to every peep. Nor do you have to defend every attempt your opponent makes to come into your territory. If they break 50 points here, you break 50 points there. The game will still be even. Be willing to give up stones for thickness or for sente.
Not even going to guess at this one.
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) ive experienced improvement as a struggle for confidence mostly. confidence in ones own reading ability and judgement of big points is necessary in order to be able to play away at crucial moments.