I've written a tutorial a while ago; I will be using some terminology in my answer and it would be best to acquaint yourself with it.

Prop (proportional) is one of the easiest to understand. All miners submit shares, and after the pool finds a block, the reward is divided proportionally among the miners based on how many shares they submitted that round.

PPS (pay per share) is also quite straightforward. Each time you submit a share, you get paid a fixed amount. As I mention in my tutorial, that amount is equal to the block reward divided by the expected number of shares that round. With PPS, the pool operator takes on a lot of risk (e.g if a block is not found in 500% the expected time, the owner is basically paying out a lot without any compensation in return), so typically they will charge a large pool fee (as a risk premium). For instance, BTC Guild charges 7.5% for PPS compared to 3% for PPLNS.

PPLNS (pay per last N shares) is similar to Prop but is designed to combat a major problem with it: pool-hoppers. Basically, since mining is a Poisson process, there's no such thing as partial credit. If a pool finds a block at 10%, it is worth just as much as a block found at 1000%, even though it took 100x less time to find it. So it is advantageous to mine only when the pool % done is low, and then switch to another pool that has a lower % done, because your expected payout per share is higher when the % done is lower. PPLNS solves this problem by making it by (roughly speaking) distributing the block reward not according to who submitted shares in that round, but who submitted submitted the last N shares (N is often twice the expected number of shares per round). I won't go into the full mathematical details here, since it's explained perfectly well by other people.