I'm not familiar with MMOs, so Raid is a bit of a foreign term to me. But I'll explain how it applies to Destiny.
A raid is a huge, cooperative challenge that requires the highest coordination and communication between the players involved to surmount it. In Destiny's case, you're given no direction and you have to be playing with friends (no randoms like in Strikes). It's up to you and your friends to figure out how to progress.
It sounds like there will be platforming challenges, puzzles, bosses, and a variety of other stages within the raid. Completion will net you some pretty awesome and specific loot apparently.
There will be a normal and hard mode. The folks that Bungie brought in seemed to struggle in completing it, so it should make for some great post-game content if it lives up to the talk.