If you've ever played WoW(or similar MMOs), here's a quick comparison:
Flashpoints = Dungeons. The max player limit is 4 people, regardless if they're players or Companions. For example, a Flashpoint group could be 4 players, 3 players + 1 Companion, 2 players + 2 Companions, etc. They're generally balanced for 4 actual players.
Heroic quests: Generally balanced for 2 players + 2 Companions, but sometimes more. (It says in the quest title how much it recommends.) These quests generally feature Elite+Champion level mobs, and are done in a non-instanced area. They reward XP, and one blue-quality item, usually.
Operations = Raids. They have 8-player and 16-player variations of the same Operation, and have 3 difficulty modes: Normal, Hard, and Nightmare. The harder the difficulty, the better the loot bosses drop. 8-player and 16-player variants share the same lockout, as far as I know.
I'm not too sure the mechanics behind the Need/Greed system. I'll see if I can dig anything up.