From the moment you see that opening you know you aren't watching a normal anime (an English song in my Japanese cartoon opening?? [they're a Japanese band though, which is pretty amazing]). Ergo Proxy is the story of...well, you aren't really sure to begin with. You're sort of thrown into the middle of things with very little back story. As it goes on you quickly piece together that the main characters are Vincent--a emigrant trying to become a "fellow citizen"--and Real--a spoiled woman that is in high standings because of her grandfather. Soon, Real stumbles onto the existence of a couple of monsters (that destroy her home) and after wards everyone is dead set on convincing her that they weren't real. And, poor Vincent, is just looking for acceptance by working hard when he gets blamed for a crime and chased by not only the police but also the monsters that attacked Real.
Any more plot than that and I would give away too much, part of the greatness of the series is watching as it all unfolds. The characters are fantastic (the other main one being a robotic little girl that, amazingly, isn't annoying but funny and lovable), the story is great and mysterious once it gets going, and somehow the show never loses that feeling that you've been thrown into something that you AND the characters have to swim through to get to the truth.
Highly recommended!