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.