OK, just in case this needs spelling out, this will contain ENORMOUS SPOILERS. It's essentially all spoilers. But if you've already finished the game, or more importantly have not, will not and still want to see what all the ruckus has been about, read on!
This isn't a definitive guide on how to achieve them. If some kind of Google magic has led you here thinking otherwise, sorry. It's just a collection of the most important ones so you can arm yourself for one of the most passionate discussions about a game's ending we've seen in a long time.
There are, apparently, sixteen different endings to the game. Or, should I say, "different", because many of them are largely identical, sharing most animated sequences and differentiated only by changes to an effect's colour palette.
The endings you receive when completing the game depend on two things: whether you destroyed the Collector Base at the end of Mass Effect 2, and what your Effective Military Strength was when setting off on the game's final mission.
Many players when reaching the end of Mass Effect 3's playable content are given two options (either destroy or control the Reapers) with how to save/change the galaxy. If you'd passed certain criteria, you were also given a third, neutral one (synergy).
Since many of the clips are almost identical, I've included just the important ones (those showing the most dramatic differences) in the gallery above.
In the first one, you'll see one of the "best" endings (really just an additional few seconds shown after a regular conclusion, at the 4:31 mark), in which it suggests that Commander Shepard somehow survives the whole ordeal. Leading to the week's most popular conspiracy theory.
The best "Destroy" ending. All Reapers are destroyed, the Earth is saved, the Normandy escapes to an alien world. Head to 4:31 to see the SHEPARD LIVES moment.
The"Synergy" ending, in which all synthetic and organic beings somehow become one. Again, the Normandy escapes, and pilot Joker now has the added benefit of being able to knock boots with EDI.
The best "Control" ending. Shepard sacrifices his body to take command of the Reapers, flying them away from Earth. The Normandy once again survives.
The worst "Destroy" ending, in which you didn't get enough forces together and made some pretty crummy decisions. Reapers are killed, Earth is destroyed, everything is super depressing. It's even suggested the crew of the Normandy are DOA.
An average "Control" ending, in which for some reason the Citadel, which explodes in just about every other ending, survives.
And here are all the endings synced together. Just for shits and giggles.