The first betting rounds
Omaha is usually played with a small and a big blind, which are posted before the cards are dealt and considered as live bets.
Four cards are dealt face down to each player and the first betting round takes place as usual.
When all players have either folded or called the last bet, the flop is dealt: three cards face up on the board. These cards are community cards, which mean that all players can use them to build the best possible five-card poker hand.
Then the second and third betting rounds take place just like in Holdem. Omaha is mostly played with fixed limit or pot limit, but no-limit Omaha also occur.
The river and the showdown
When the fifth board card has been dealt (the river), the last betting round takes place.
If more than one player remains in the hand when the betting is over, there's a showdown. The player who put in the last bet or raise and was called shows his or her cards first. The other players then show their cards in clockwise order. If they don't want to compete for the pot, they can fold their hands without showing.
The player who can build the best possible five card hand using two hole cards and three board cards wins the pot. If several players have the same hand, they split the pot equally
Note that the cards that are not included in the five card hand are neglected. They play no role when deciding which hand won. In the same way, different suits do not have different value.
Some peculiarities of Omaha
With four hole cards instead of two, you may think there would be twice as many two-card combinations as in holdem, but there's actually six times as many. This is what makes Omaha very different from Holdem, after all.
With a hand like A-K-Q-J, you can form six decent two-card hands: AK, AQ. AJ, KQ, KJ and QJ. Of course, this improves your chances of hitting the flop very much compared to Holdem.
But your opponents' chances also improve. In Omaha, the poker hands that people show down are usually much better than in Holdem, on average. You need to adjust to this. Stop moving all in with bottom two pair, and such.
You also need to understand that if one of your hole cards doesn't connect with the others (a hanger), your starting hand is much weaker. A hand like A-K-Q-7 has only three decent two-card hands, AK-AQ-KQ, so in a sense it's half as strong as A-K-Q-J.
Since your opponents will often have many kinds of hands working on the flop, if you hit it good you usually need to protect your hand. Slowplaying is often much more dangerous in Omaha than in Holdem.