Legacy is sort of a collection of all your characters.
When you finish chapter 1 with one character you get to choose a surname for him. That becomes your Legacy name for that server. You can choose to use it as a surname for other characters on the server (don't have to). You then can set up relationships between them. You can set up friends, comrades, or a family tree. Doing so will allow you to unlock skills as you level up your legacy. For example, your main is a Smug and you level him to 50 and gain a bunch of Legacy points. Make a Trooper level her up and make her Smug's wife. By then your legacy should be a pretty high level, so you make your next character a Consular. Set her as the others' daughter and she'll "inherit" some of their skills/abilities--she can kick guys in the crotch for example.
That's a pretty simple example, and they've said they going to gradually add functionality to the system so that (eventually) players will be able to put together some rather complex relationship chains.
Of course, a lot of that is more for RP purposes.
Leveling your Legacy unlocks other stuff, too, like being able to play certain races with classes that wouldn't normally have access to them. There are a few special buffs you can unlock along with gear and different speeders etc.
A lot of this stuff you can just pay for once you meet the proper legacy level even if you don't meet all the requirements needed to otherwise unlock it. Though it's all really expensive--they obviously intend it to be a huge gold sink.