Chess strength is a funny thing. It's hard to define precisely, so we rely on rating systems (primarily Elo-based) as a proxy statistic for it. Yet clearly there must be something substantive behind the explanation for why players have particular levels of strength, so we can talk about things like "master" and "expert" versus "Class B" and "Class C" players. (I am currently a Class B player, per the USCF scale above.)
There are some helpful, if not necessarily definitive, attempts at providing "roadmaps" or the like to chess knowledge at each level. Here is one posted at Chesstempo. You can also infer what knowledge is considered standard to have at beginner, intermediate and expert levels from resources such as the Chess.com study plans. Another approach is defining specific characteristics and skills that set the higher levels apart from others, as done by GM Andy Soltis in What it Takes to Become a Chess Master. Soltis' book I think gets at some crucial concepts, including how masters are able to much better understand (and apply) things like compensation for sacrificed material in the absence of concrete winning tactics. This includes "positional" sacrifices of the exchange or a pawn or two, where there is no combination on the board, yet the master understands that in the long term, their chances are better (or at least as good) as before the sacrifice.
All of the above approaches to explaining playing strength have their uses, but one of the conundrums of chess strength is that it often does not reflect the extent of a player's knowledge, at least on a one-to-one basis. Some things are directly correlated with your ability to win; knowing basic mates and mating patterns are fundamental to success. Others are helpful, but not 100% required (for example how to play the Philidor and Lucena ending positions). Finally, some pieces of chess knowledge have a very low (yet non-zero) percentage chance of ever being directly helpful (such as knowing the K+B+N v K mate). Soltis' work outlining particular crucial areas of skill I think comes closer to a cognitive approach, rather than simply giving a list of "must knows", but of course this approach is necessarily subjective, rather than rigorously scientific.
Similar to Soltis' approach, but on a more practical level for the improving Class player, I'd like to document some phenomena that appear as concrete indicators of improving chess strength. It's not an exhaustive list, but I think it is useful to share. I make no claim to having originated any of these ideas, but in reading widely and through analyzing my own games I have been struck by how important some of these are when they appear, and I have personally experienced all of them at one time or another. Naturally there are many more improvements yet to come...
I've arranged the below phenomena in what seems to me to be a logical progression from ones based on more concrete/tactical/conscious considerations, to those that are more cognitive/strategic/unconscious in nature. Again, these are meant to be taken as positive signposts you may see on the road to mastery, whenever you observe them in your games. I think it's important for improving players to explicitly recognize the positives in their game, along with the many errors and negatives, otherwise chess can start becoming toxic and demotivational. We (including professional players) will always fall short of perfection, but that's to be expected, and therefore not overly lamented.
- I hesitate to start with this one because of its obviousness, but blunders (making less of them) really is fundamental. If you keep having the same frequency of game-ending blunders (i.e. losing a piece or overlooking mating threats) over time, then by definition you will make little progress. Having at least a basic blundercheck thinking process is necessary; over time, it hopefully will become more and more unconscious (but still necessary). One practical observation I've read before, and tend to agree with based on personal observation, is that by Class B level the majority of games are no longer decided by blunders, and by Class A only a very few are. In other words, you more often have to beat your opponent, rather than just waiting for them to beat themselves.
- When analyzing your games, you have an increasing number of "!" annotations. This is related to the above indicator on blunders (the "?" moves), but it is not simply the elimination of errors, but rather a demonstrable ability to find the key (sometimes only) move that gives you a breakthrough in the position. You can always give yourself a "!" if you're feeling generous, but it's worth more when coming from an unbiased annotator, which in many cases will mean your computer engine. Although there are some pitfalls of computer analysis, engines can almost always correctly identify the standout strong moves ("!") as well as the blunders.
- Something a little less obvious, but a definite sign of improvement in strength and sophistication, is spotting and deliberately using intermediate moves when conceiving and calculating move sequences. One thing I have seen repeatedly in my game analyses is that I (or my opponent) often may have a good idea, but it is not executed to the full extent of its power, whether it may be a strategic pawn break or a particular tactic that becomes much more effective with the insertion of an intermediate move. I think that learning how to keep searching for more effective moves after finding a good idea is the key, including taking a particular idea and then looking creatively at the different possibilities of how and when it could be most effectively played. Annotated Game #171 has some good examples of effective intermediate moves.
- As part of your evaluation of the position, you start naturally "thinking in squares" rather than just about the pieces and threats to them. Weak squares in your opponent's camp become magnets for tactical and strategic ideas; one common example is if f2/f7 become underprotected next to a castled king. Excellent squares should also suggest themselves (see below) for your pieces - for example leading to the repositioning of a knight, perhaps even using a tactic, to get it to a dominating outpost on the opponent's side of the board. Defending your own weak squares and realizing when they might be created (after a pawn push, when a key piece moves away) is also very important, the more so the higher level you get. (See this example commentary game.) Going back to the thinking process, Botvinnik suggested thinking on your opponent's time about positional considerations rather than calculating variations - squares are a big part of this process.
- Allied to the automatic recognition of "thinking in squares" is the development of a more automatic/unconscious visualization skill, in other words the ability to picture future board positions in your mind. I have found that this grows naturally in the context of a consistently applied training/study plan, as long as you sometimes move the pieces in your head rather than always on the board. One good example of doing this is when variations are given in an annotated game and you visualize them, rather than playing them out physically. This can also be practiced in reading chess books without a board, starting with ones that have plentiful diagrams (such as Logical Chess: Move by Move). At a basic level, you should not need to have the board coordinates (A-H, 1-8) printed on the board for you to immediately identify a square. The next stage is being able to mentally picture (away from a board) the color of a particular square (d4 is black, b7 is white, etc.) quickly; if you can't do this, then your internal board sight is not functioning on an unconscious level and the need to process this consciously will slow you down. Finally, being able to play a game "blindfold" (without sight of the board, whether or not you actually are wearing a blindfold) is more than a parlor trick, it is a sign that you have strong visualization skills that will strengthen your calculation and evaluation of variations. Blindfold chess skill is again not a one-to-one correspondence with overall chess skill - not everyone can be Timur Gareev and he is not at the super-GM level (yet) - but it is a strong indicator of your general strength level.
- While the above are largely conscious mental efforts related to the process of calculation - with visualization being, I would argue, also a partially unconscious function - one unconscious-origin phenomenon is when a move "suggests itself" without you doing any calculation at all. At a certain level this phenomenon is related to what we call "natural moves" - which to a lower strength player do not necessarily appear natural at all. Geometrically these can typically be defined as when an individual piece can be moved to a square where it will have maximum influence across the board, particularly into the opponent's camp. More sophisticated versions of this idea occur when we have a mental library of successful positional patterns built up, so we make an instant comparison to what we have seen work before. To quote Magnus Carlsen: "Of course, analysis can sometimes give more accurate results than intuition but usually it’s just a lot of work. I normally do what my intuition tells me to do. Most of the time spent thinking is just to double-check."
- Finally is an observation that I once read by a GM (the source regrettably does not occur to me right now, maybe Yermolinsky?) who noted that when you gain strength you do not in fact feel any stronger yourself; rather, your opponents start seeming weaker (in the same Elo range as you). This is naturally a largely unconscious impression, but I believe it's a valid one and probably one of the most powerful indicators that you have in fact shifted significantly past a previous milestone in chess strength.