Many times during any given game, one asks themselves... "Should I defend, or attack? Are the tactics advantageous now, or do I need to create another threat/protect another piece? Can I snatch up a pawn or two without weakening my position and losing the material advantage?" However, I don't think most people (certainly I didn't used to) think of these things in terms of priorities. There are two priorities that are always highest on the list.
1) Protect your king. If checkmate can be forced on you, the game is over. This is always first priority.
2) Checkmate the other king. This, if it can be forced, can also save you the trouble of all your other problems.
After these two, all the other priorities kind of jumble together. Material advantage is a priority, but so is the initiative. Forcing pieces onto poor squares is a priority, but your pieces also have ideal destinations.
I have found a few examples of these types of decisions. The first example is a game of my own, which ultimately was won because my opponent did not keep priority number one in the forefront of his mind. Material remains even through the entire game; but, he becomes blinded by what he sees as a checkmating attack (priority number 2), and forgets about priority number 1: defending.
The threat of Qa3 in that game really does come out of nowhere. I feel this only emphasizes the idea that one must always be on the lookout for threats to the king (once again: priority number 1).
The next four examples are chosen because they share one trait: in each, one side's queen takes the other's unprotected g-pawn. In three, this action is safe: in one, it is not. The motive for this is normally to win a pawn, and therefore gain an advantage in material. In each case I will explore if this is worth it.
The first is simple opening theory in the Two Knight's Defense. White seems to think that he can obtain a quick material advantage. Black is allowed to develop enough that Black in fact wins material (or checkmates).
Black could afford to take the g-pawn because of the threat of checkmate. Now, if White takes priority #1 into account and simply plays 6. c3, he has defended and is absolutely fine. But overhastiness (and a greed for material) play against him.
The next game is one that also works against the player trying to swipe material. The game is not decisive, but in very quick timed games proper defenses are often missed.
The next game shows when taking the g-pawn is obviously NOT advantageous. I believe Nimzowitsch's opponent in this game was an amateur, and therefore must have been disproportianately tempted by the win of a pawn... But Nimzowitsch's advantage in development and excellent piece coordination punish swiftly and fiercely. And let's not forget, even if his piece coordination was a fraction of what it was... taking the g-pawn opens the g-file nine times out of ten. Even that compensation is sometimes enough to make one consider jumping on a material advantage.
Of course, one of the best players in the world vs. an amateur would normally prove to be an exceptional example of simple principles in action.
The next game, in contrast, is between two rather strong players. And where the first four examples showed very clear comparisons... where the material gain was worth it and where it was not... this last game shows White's capture on the kingside very much countered by Black's advantage on the queenside. This is the normal way of things... when prioritizing is unclear. I think the reason White's g-pawn capture is advantageous here is because of the proximity of the Black king, and also the wonderfully multi-purpose e-pawn; contrast this to Black's queenside push, which, while impressive in its own right, creates threats less serious than White's (checkmate and promotion).
And so ends my ridiculously long reflection on what important aspects to consider. A quick recap and list of some of the priorities:
1) Defend yourself from checkmate.
2) Checkmate the opponent
3) Maintain a material advantage
And, of course, positional moves, pawn structure, tactical exchanges, protecting pieces, moving the opponent's pieces, and also initiative, which I believe welds itself amongst all of these.