What modes of multiplayer are present in Mass Effect 3?

The multiplayer isn't really story-based. It's a wave-based survival mode (think Gears of War "Horde") with 10 enemy waves + an "extraction" wave. There are 3 levels of difficulty (Gold, Silver & Bronze), and several playable maps and enemy types.

It's designed to be played with 4 players and has a baseball card style unlock system where you can buy packs of items. These items include weapons, weapon upgrades, health packs, and playable characters.

In the demo, you purchased these packs with in-game currency earned by completing objectives. In the final game, you can also purchase these packs with real money.

Ultimately, the multiplayer is just a fun diversion based on the combat of the main campaign. In addition, the multiplayer can affect your campaign story by improving your "Galactic Readiness" score, but it's not required to get the best ending in single player.

The multiplayer consists of you and up to four teammates engaging in survival-style gameplay to complete 11 waves of increasing difficulty before being wiped out. Most of the rounds involve you and your teammates killing all of the enemies, but occasionally there will be an objective wave which involves killing a set number of high value targets, hacking a data terminal or disarming several objectives.

Teamwork is essential during multiplayer, as players are capable of rescuing downed teammates if they can get to them before an enemy finishes them off or they bleed out. There are six different playable classes with many different abilities, allowing a great deal of variability in tactics. For example, some waves feature enemies with large shields that are very difficult to attack from the front, but from this question, you can see that there are quite a few different approaches you can take to dealing with them depending on the abilities that you and your teammates have available.