Pot Odds:
When you bet (or call a bet) you are, of course, trying to win the money that is already in the pot. How often do you have to win to make this profitable? Clearly not every time - if it costs you 10 to bet (or call) and there is 100 in the pot, then you'd be able to be wrong 9 times out of 10 and still break even. This is the essence of the pot odds: You're paying a fraction to win a larger sum. If you're more likely to win than you have to pay, then your bet/call is a winning move in the long run.

Let's try one of the standard examples for pot odds: The flush draw.

First you need to consider your odds on hitting the winning hand. In the case of a flush draw on the turn in hold'em, you're getting about 4-1 (actually 37-9, since there are 37 cards that will "miss" you, and 9 that will give you the flush, but 4-1 is a good enough approximation) if the flush will be the best hand. This means that the pot odds need to be 4-1 or better in order to make your draw profitable. For instance, if your pot odds are 3-1 (paying 10 to win 30) you would get this Expected Value calculation:

(-$10 x 37/46) + ($30 x 9/46) = -$8.04 + $5.86 = -$2.17


What does this mean? It means that if there's only $30 in the pot and you have to pay $10 to win it, you'll lose on average a little over $2 every time you do it. Not a good thing.

What if the pot was $50?
(-$10 x 37/46) + ($50 x 9/46) = -$8.04 + $9.78 = $1.74

Here, you average $1.74 profit for every call you make.

Outs and Odds:
An easy way to work out your pot odds is to count your outs (cards that improve your hand) and do this simple calculation.
Eg. You have a flush draw on the flop giving you 9 outs. For one card (the turn) multiply the outs by 2 then add 1, so 9x2+1=19. The answer 19 gives you the percentage chance of hitting so you have a 19% chance of making your flush on the turn.
That?s 19% in favour and 81% not in favour. This gives approximate pot odds of 81-19 or to simplify 4-1.
For two cards (the turn and the river) multiply your outs by 4, so for the flush draw with 9 outs you have a 9x4=36% chance of hitting your flush.
That?s 36% in favour and 64% not in favour. This gives approximate pot odds of 64-36 or to simplify 1.8-1
Memorising these might help:
20 outs = 0.5 to 1 (1 card) and 1.3 to 1 (2 cards)
19 outs = 0.5 to 1 (1 card) and 1.5 to 1 (2 cards)
18 outs = 0.6 to 1 (1 card) and 1.6 to 1 (2 cards)
17 outs = 0.7 to 1 (1 card) and 1.8 to 1 (2 cards)
16 outs = 0.8 to 1 (1 card) and 1.9 to 1 (2 cards)
15 outs = 0.8 to 1 (1 card) and 2.1 to 1 (2 cards)
14 outs = 1 to 1 (1 card) and 2.4 to 1 (2 cards)
13 outs = 1.1 to 1 (1 card) and 2.6 to 1 (2 cards)
12 outs = 1.2 to 1 (1 card) and 3 to 1 (2 cards)
11 outs = 1.4 to 1 (1 card) and 3.3 to 1 (2 cards)
10 outs = 1.6 to 1 (1 card) and 3.7 to 1 (2 cards)
9 outs = 1.9 to 1 (1 card) and 4 to 1 (2 cards)
8 outs = 2.2 to 1 (1 card) and 5 to 1 (2 cards)
7 outs = 2.6 to 1 (1 card) and 6 to 1 (2 cards)
6 outs = 3 to 1 (1 card) and 7 to 1 (2 cards)
5 outs = 4 to 1 (1 card) and 8 to 1 (2 cards)
4 outs = 5 to 1 (1 card) and 11 to 1 (2 cards)
3 outs = 7 to 1 (1 card) and 15 to 1 (2 cards)
2 outs = 11 to 1 (1 card) and 23 to 1 (2 cards)
1 outs = 23 to 1 (1 card) and 46 to 1 (2 cards)