What are Outs in Texas Hold'em Poker?
If there are one or more cards still to be dealt and you think your hand is currently inferior to an opponent's hand, the "outs" are cards not yet revealed to you that will improve your hand from a losing to a winning position.
So, for example:
You have an Ace-high. Based on his betting in this hand or earlier in the game, gut instinct, the fact you've seen his cards, whatever, you think your opponent has a pair of Kings. So you think are currently in a losing position.
However, if you hit an Ace on the Turn, you will have a pair of Aces, which beats a pair of Kings. There are four Aces in a deck, you have one of them, so there are three unaccounted for, so you have three outs.
A more complicated example:
You have a pair of Queens, and a four-flush in Hearts. You think your opponent has a pair of Kings, so you think are currently in a losing position.
However, there a number of cards that will improve your hand sufficiently to beat a pair of Kings. Specifically:
So, you have 3+2+9 = 14 outs. Of course, if any of these happens, your opponent still has at least five outs - the two as yet unseen Kings, which would give him three Kings, and the three cards of the same rank as his second hole card, which would give him two pairs.
The reason this is of interest to you is that it enables you to calculate what chance you have of improving your current hand to a winning hand, which in turn enables you to calculate how much you can bet (as a percentage of the pot) and still have a positive expected value for that bet.
See here for a more comprehensive explanation.
To expand on @lilserf's answer... An out is a mathematical probability that one player has to win the hand depending on the situation, and they are always used, not just mostly in all-in situations.
You have to treat them as probabilities because there is always the chance for runner-runner cards which produce a winning hand. For instance, if you have three to a flush, and making a flush is your only opportunity to win the hand on the flop, you still have outs even though there isn't a single card in the deck that will put you in the lead on the turn.
"Outs" are the cards left in the deck that could possibly make you win. Usually these are discussed most when players are all-in and the cards have been revealed.
If the only way you can win is to draw one of the 3 Aces left in the deck, you have 3 Outs.