It's kind of hard to classify. It's generally a drama, but it also has a lot of comedy. It is a romance, but it also has some elements of metaphysics and science fiction.
Well, the story is that Joel (Jim Carrey) is a very introverted cartoonist who meets a very outgoing, borderline crazy girl, Clementine (Kate Winslet). They date for months, but they eventually have a really ugly breakup. Clementine hears about a medical procedure that allows her to completely erase Joel from her memory. She does so, and when Joel finds out about it, he decides to have the procedure done as well, mostly out of revenge for her doing it to him. However, once he begins to undergo the procedure he begins to realize how much he truly loves her. The majority of the film takes place inside Joel's head, through his memories with Clementine, and as he sees how much he loves her, he decides he doesn't want to erase her memory, but doesn't know how to stop the procedure. And, meanwhile, there's another plot going on involving the people who are actually doing the procedure on Joel (Mark Ruffallo, Elijah Wood, Kirsten Dunst) and the doctor supervising the whole thing (Tom Wilkinson) that ends up intersecting with the Joel/Clementine plot in ways I don't want to spoil.
It's less complicated than I made it sound, but like I said, you do have to pay attention. Because the procedure works from the most recent memory back, most of the scenes play in reverse order, so you really have to keep track of everything.
Overall, though, it's a romance. There you go, it's a very quirky romance. It's by the same guy that wrote Being John Malkovich and Adaptation, so if you've seen those, you know what to expect.
If you have a pretty fast connection, you can watch the trailers
here.