Foreword: if you are an absolute beginner as concerns poker, i strongly suggest that you first read my article on Texas Hold'em rules.
Indeed, the Texas HE article would tell you the specific rules for HE but also the basics that you have to know in order to play any poker variant (like seats and dealer drawing, the three major betting structures ...). I won't duplicate this kind of info here.

Omaha (also called Omaha High, Omaha High/Low being another game) is one of the most popular poker games, mostly in Europe where it's kinda more popular than stud.
Its structure is almost the same as Texas HE, which makes it handy for beginners to start learning Texas and then discovering they are able to play Omaha too in their first session.

To make it short, let's say that the two games have the same structure, except for these 3 differences:

=>at the beginning of a hand, you are dealt 4 hole cards instead of 2
=>at showdown, you will use 2 AND ONLY 2 hole cards and 3 community cards to make your best 5 cards poker hand. With your 4 hole cards, you will have C(4,2)=6 ways to choose the 2 cards you will use.
=>Omaha is played with a Pot Limit (PLO) or Limit (LO) betting structure. There's a nice balance in this game between made hands (like sets) and drawing hands (like straight and flush draws) which makes a pot limit structure more appropriate than a no limit structure.

A few screenshots from an online pokerroom:



Blinds are posted and cards are dealt:

=>the dealer deals 4 cards to each player, one at a time, clockwise, starting with the SB and ending with button (on the previous picture, you can see that the button is the last to get its third hole card)
=>once hole cards are dealt, a first betting round begins

In this hand, before cards are dealt, abedm and aDutchboy1 have posted 0.5 and 1$ blinds. But, harri589 (seat 1) have posted a 1$ blind too. Why? The rule is that, when you seat at a poker table, in order to play you have to:

=>wait to be in the big blind
=>or, if you want to play right now, you will have to post a big blind, which harri589 has elected to do. This forced bet is considered as a live bet and has the same rights than a big blind, which means that, if the pot isn't raised when it comes to you the first time (you don't keep this opportunity on later rounds), you can raise it (looks like you are raising your own bet)

If you aren't a new player but have missed a blind (because you left the table, took some time to rebuy etc...) then, to re-enter the game, you will have to wait for the big blind or post a live bet (equal to the big blind) and a dead bet (equal to the small blind). The dead bet is placed at the middle of the table and isn't counted in your bet. That means that, if the pot is raised to you (say it's a minimum raise to 2$), you will have to call 1$ (you would call 0.5$ if the whole 1.5$ forced bet was considered as a live bet). Same thing if you raise, your dead bet won't be counted in your raise.
This rule is obvioulsy intended to avoid that smart asses leave the game when the big blind comes to them and then sit in 2 minutes later when they are on the button, so that they would never pay the blinds.

Then, the first player left to the big blind (zigzagdro) folds his hand, camo3 limps (=calls the big blind), all following players fold to harri589 who merely checks (he didn't used his right to raise the unraised pot).
Button (newtonfruit) raises to the maximum, which is, as it is a Pot Limit table, 3x1 (bet to call) + 2.5 (pot before the previous caller)=5.5$.
It's a particular situation where the button doesn't need a strong hand to make that move. Among his 4 potential opponents, 3 are blinds, only camo3 has decided to play this hand and he only limped. Unless the blinds are somewhat loose players, button has to think it's a good spot to take down that pot.



Dealer burns a card and places 3 community cards (called the flop), face up, at the middle of the table.
Second betting round.
Here the first 3 players check, fourth (gusar99) folds. With that type of flop, if he has very small cards with no flush draw, he can rightfully think he doesn't stand a chance to win that hand.
Fifth and last player fires a 3 quarters of the pot bet.
With 3 opponents still in the hand, you will quickly learn that it's not wise to try to bluff in this spot. There are far too many cards out, somebody's gonna call you. So, if el_dudarino is a solid player, he's got to have some kind of hand.
In fact he had a pair of aces and will be beat by someone who flopped the lower straight and a small flush draw which will hit (8 high flush i think).

Once this second betting round is over, and provided at least two players are still in the hand, dealer burns a card and places a fourth community card (the turn), face up, at the middle of the table.



A third round of betting.
Here two players are still in the hand when the turn is dealt.
tallaferro check/folds to mojo786's 10$ bet (as it is Pot Limit, any bet between the big blind, 1$, and the pot, 18.05$, was possible).



Still two players.
The first (PKKSTANLEY) fires a 50$ bet, the second (melindsch) calls all in (33.50$)
You can't see it because I made the screen capture a tad too early but, the hands were:
PKKSTANLEY: his final hand being :
melindsch: her final hand being:
PKKSTANLEY's 3s full of aces is beat by melindsch's kings full of threes. Had the pot not been raised preflop, a three (four of a kind) might have won that pot.
This hand is a good illustration of the fact that hands in Omaha are stronger that in HE. When there are lots of action in a multiway pot, don't go in deep money without having a very big hand, preferably the nut hand or the nut draw. It's almost sure that somebody will.
That is why PLO is called the "Big Money Game". There are so many clashes between stong made hands or strong hands and strong draws that pots are in average bigger than in HE.

Note on the 2 cards constraint: The fact that you have to take MANDATORILY 2 cards from your starting hand make beginners do mistakes. That makes Omaha something different than a mere four cards Texas.
Example: Board reads:

Your starting hand:

Your opponent's: