Pot odds are quite simply the odds offered by the pot, that is, the ratio of money you will win if you win the pot compared to the amount you would have to bet to remain in the pot.
For instance, after the turn in hold'em there are 7 big bets in the pot when the betting gets to you. You have four cards to the nut flush. You have to put in 1 bet to continue, so we say the pot is giving you 7:1 (pronounced "7 to 1") odds. Compare that to your approximately 4:1 chance to make your flush on the next card. Since your chance of making the flush is better than the odds being given by the pot, you should call. What if there had been very little betting and there were only 3 big bets in the pot when the action got to you? The pot would be giving you 3:1 odds versus 4:1 to make your flush. Your odds are not better than the pot odds, thus the correct action is to fold.
The same calculations can help you decide if you should call after all the cards are out. Instead of comparing the pot odds to the odds of making your hand you have to compare the pot odds to the odds that your hand is good.
Some common misunderstandings
This is raw content, possibly from a cross-post licensed by its author, that needs to be cleaned up.
Q: Pot odds depend on knowing how many outs remain in the deck. Don't I need to know what cards were folded to be able to calculate my outs?
Q: Should I try to esimate how many of my outs are in the opponents' hands?
This way of looking at it is far better than the previous question. It takes into account the EV for the number of flush cards in folded or live hands. It's not wrong, just adding needless complexity that doesn't affect your calculation at all.
You have two spades, two spades are on the board, one card is to come. Just for kicks, let's try accounting for the spades in the nine unseen hands (whether mucked or live). There are 18 cards in those hands, which is 18/46 of the remaining deck. So the EV for number of spades in those hands is (18/46) * 9, or 3.52.
So we expect that 9 - 3.52 => 5.48 spades are left in the remaining 28 cards (stub plus burn cards). Hence there are 22.52 non spades left on average. Our odds against catching a spade on the river are 22.52 : 5.48 => 4.11 : 1. This happens to be precisely the same as 37 : 9.
The proportion of spades in the pool of unseen cards isn't changing just because those cards were or weren't dealt out. None of those cards in someone else's hand is substantially more or less likely to be a spade. (That's slightly untrue. One or more of your live opponents might also have a spade draw. But this possibility is negligible.)
Note that this only applies to unseen cards. If the dealer accidentally exposed the 5 ♦ and burned it, then it would affect your calculation, because it's obviously no longer eligible to come up on the river.
Since it's easier to consider all 46 cards to be one pool of cards, and doesn't change the result one iota, that's how most people choose to calculate it.
- implied odds
- effective odds
- correctly calculating outs, especially discounted outs that aren't certain to win the pot.
- Possibilities of raises behind you.
TODO: various examples as appropriate.