Analysis is harder to analyse, as it seems to consist of three sub-skills: calculation, evaluation and pruning. Calculation is the "if they do this, then I do that..." kind of thinking, which requires a systematic approach that does not miss any key moves. Evaluation is the assignment of a value to the position reached when calculating a line and comparing it with positions reached by calculating alternative lines.
So what's this "pruning"? It turns out to be a key skill that distinguishes the master from the amateur. Contrary to the naive belief that a master calculates much more than an amateur, in a sense the opposite is the case. The master prunes the tree of analysis by not investigating moves that are clearly inferior. Based on their experience and intuition, the master has a good feel for which moves need to be investigated at some depth, and which ones merely cloud the picture. This saves the master from being overwhelmed by the myriad possibilities of a chess position. By contrast, the beginner gets lost because they try to cover all the possibilities, which is an impossible task.
Of course, there are other skills involved in chess, such as planning, the ability to adapt to changing situations, risk management, and pattern recognition. The first two of these are the creative aspects of playing chess. Planning is the creation of tactics and strategies. Without this your game will drift like a leaf in a gutter, and probably go down the drain. Flexibility is equally important, as no matter how clever your plan, it is always possible that your opponent has an idea that may invalidate yours. If your opponent is at a similar level of skill then the execution of your plan is more than likely to strike a hitch, requiring a re-think.
Risk management is needed because to win a game we often have to take risks, such as engaging in a speculative attack. In many cases we do not know whether a sacrificial attack is sound or not, nor whether we should be focusing on attack or defence, as this depends on an evaluation of whose attack is more dangerous. Once the game has progressed we need to evaluate whether we are playing for a win or a draw, based on our assessment of our position.
Pattern recognition is often subliminal but it is another skill that clearly distinguishes the master from the amateur. Every master has played thousands of games of chess, and analysed many of these. The patterns from these games are templates in their long-term memory. They enable the master to see the essential features of a position, including what sort of lines may lead to a win.
There is one more essential skill: prophylaxis. This is the ability to see the game from your opponent's point of view, ie to create tactics and strategies on their behalf, as well as how to frustrate them.
Since all human players make mistakes, one more faculty is needed: an active eye for errors while we apply the skills mentioned above. At the least, we must not miss a viable move in response to the next move we plan to make.
There are at least three quite distinct kinds of chess: over-the-board, turn-based, and blitz. Classical chess, ie over-the-board play, requires all the above skills. Turn-based chess (where you have one to three days to make a move), is similar except that one can avoid visualising altogether, due to the analysis board. The only thing I hate about turn-based chess is that when I make a blunder I can't blame it on lack of time or my inability to see the board five moves ahead! Blitz or lightning is quite a different game. Between weak players it is largely a game of chance - who is quicker to make the biggest blunder? Between strong players it requires the ability to play intuitively, ie with severely truncated analysis trees.
A cynic might add that another modern chess skill is using software assistance during a game. Needless to say, no-one reading this would do such a thing! I certainly have never done so and never will. The person who resorts to cheating may well win the game, but in truth they are cheating themselves. A win arrived at by cheating is actually an admission of weakness and can only sap one's inner strength.