I notice golf players often miss a long putt by a small amount so that there's only a few inches remaining. In this situation they perform what I believe is called a "tap-in" and are allowed to take that shot immediately, rather than waiting for their opponent to putt. Typically when a golfer does this, they put their left foot forward (for right-handers) somewhat awkwardly. The stance is clearly unusual. Are there any rules that dictate how a "tap-in" must be performed? Or is the unusual stance a tradition?
This usually happens when the ball is very close (just a few inches) and the player is almost guaranteed (99+%) to make it. In non-professional play, players will sometimes just pick up the ball without tapping it in, but in professional play the ball must go into the cup, no matter how close it is.
As gbianchi says in the comments, the player who is farthest away is usually the next to go (Rule 10-1b), but he's not entirely correct in that someone closer is not allowed to go first. The relevant rules are 22-1 and 22-2.
Rule 22-1 and 22-2 allow for any player lifting any ball (lifting means you remove it but replace it in exactly the same spot after the other player shoots) if the ball may assist another player (Rule 22-1) or the shooting player having another ball lifted if it will interfere with his/her own shot (Rule 22-2). They both have the same clause in them:
"In stroke play, a player required to lift his ball may play first rather than lift the ball."
So for a "tap in" situation, usually the ball is so close to the hole that it may interfere with another player's putt, so the player who has the "tap in" can choose to putt it instead of lifting it. Note that this does not require the permission of the other player - on the putting green a person may lift their ball without someone else asking (this is not true off the green - see Note 1 to Rule 22-2).
Now as to the stance, the relevant rule is Rule 16-1e. It says:
"e. Standing Astride or on Line of Putt The player must not make a stroke on the putting green from a stance astride, or with either foot touching, the line of putt or an extension of that line behind the ball.
Exception: There is no penalty if the stance is inadvertently taken on or astride the line of putt (or an extension of that line behind the ball) or is taken to avoid standing on another player’s line of putt or prospective line of putt."
So because a player cannot stand on another player's putting line, they usually have to stand farther away from a "tap in" than they normally would and lean over to it.
Source: USGA Rules - http://www.usga.org/Rule-Books/Rules-of-Golf/Rule-01/
Basically, the applicable rules for stroke play in this situation are 10-2, 22-1 and 22-2:
So, it's custom to play out of turn in order to "clean up" a ball laying near enough to a hole that the player can be expected to hole out.
Whether lifting or playing a ball close to the hole, players have to be careful not to disturb the line of play of other golfers; a soft green will retain marks from the golfer's spikes which can interrupt a ball's path, so golfers avoid placing their feet along the probable line of putt of another player. This can lead to some funky stances for tap-ins, which are specifically allowed as exceptions to Rule 16-1 (which otherwise prohibits a player taking a stance standing astride or with either foot on their line of putt) because they are taken specifically to avoid another player's line of putt.
As Duncan says, in casual play, players whose balls are within some certain distance can, on approval by another player, pick up their ball and score the hole as if they would have holed out with their next stroke. It's called a "gimme", and it speeds casual play but it's technically a disqualification from a tournament for leaving a hole without having holed out.