After you've gotten your initial cards dealt to you in a poker hand, you usually need to draw or hit certain cards to make your hand a winner. Outs are the all the possible cards you could get that would improve your hand.
Most commonly, you'll be talking about how many outs you have after the flop in Texas Hold'em.
For example, if you have the Ace and Jack of Diamonds in the hole and the flop is Queen of Diamonds, Ten of Clubs, and Three of Diamonds, you have a lot of outs. First, any Diamond that shows up on the turn or river will give you a flush. Since there are 4 diamonds already showing and 9 left in the deck, that's nine outs. Additionally, any King will give you an Ace-high straight. Since you've already counted the King of Diamonds when you counted diamonds, just count the three remaining Kings, so that's already a total of 12 cards in the deck that will likely give you the winning hand, or 12 outs.
Some common scenarios in Texas Hold'em:
If after the flop you have:
Four cards to a flush you have 9 outs.
Four cards to an open-ended straight (i.e. 3-4-5-6) you have 8 outs.
Four cards to a inside straight (i.e. 3-4-6-7) you have 4 outs.
If you need to pair one of your hole cards, you have 3 outs.
The more outs you have, the better your odds are of improving and the more likely it is you will hit one of those needed cards and win the hand!