Brazilian Jutjitsu (avoid it... not efficient unless GraceJutjitsu but great community and tournaments)
It's the same thing. The Gracies invented BJJ. They have a chain of Gracie-branded gyms.
As to the effectiveness of BJJ in a real fight, you're supposed fall down like this:
And then say, "Now jump on me, and meet your doom!" I actually did that in an mma fight once. I was getting the crap beat out of me and the ref was about to call it. But I rolled back like that and the moron jumped on me. Never jump on a jiu-jitsu guy. I caught him in an oma plata, and instead of finishing the move, I pinned him down and beat on his ribs like I was tenderizing meat. On the tape later, I counted a dozen blows by sound alone, heard over a thousand screaming people. I could hear girls in the audience going "ew! ew! eww! gross!" It was the sound of the beating that was grossing them out.
As far as real fights go, BJJ works when you are losing. If someone tackles you, sits on your chest, and starts pummeling you, then you need to know how to get away, or you're going to die. I actually know a guy who died that way. A mount escape would have saved his life.
There's been a lot of sosuave threads debating the best martial art. Generally speaking, you want to learn a striking art and a grappling art. Beyond that, the place you choose to train should be influenced by the quality of instruction you have available. If it were me, I would look for a gym that produces real fighters - boxing, mma, or kickboxing. If it's BJJ, their guys should be winning medals at tournaments like NAGA and Pan Ams.
Some martial arts do not have any formal competitions, and those are the easiest ones for the sensai to be a fraud. I'm not saying they are all frauds, but if the students don't ever enter competitions, then you can't tell if they are any good or not.