Elmer Bernstein - One of the greats I'm surprised nobody's mentioned. The Ten Commandments, To Kill A Mockingbird, The Man With The Golden Arm...he's been composing for film for over 50 years, and still was as of 2002 with Far From Heaven.

Leonard Bernstein - OK, two of his "scores" were originally done for Broadway shows, but he adapted them to the screen: West Side Story and On The Town. He also scored On The Waterfront...one of the few bona fide "serious" composers that I know of to have done work in film.

1. Hans Zimmer. Hans' score to Pirates: At World's End brought me fully into the world of film scores. I then started exploring, and found Gladiator, The Lion King, and others which I can't stop listening to. He might be a little repetitive sometimes, but everything of his that's outside of that box is really great work.
2. James Newton Howard. Some of his work is just breathtaking. I bought King Kong a little while ago, and some of the themes are just fantastic (central park). And he composed it in little over a month (!!). He's particularly good with the emotional pieces.
3. Dario Marianelli. I really enjoyed Pride & Prejudice, and I bought Atonement with high expectations. They were certainly met, and more. I, however, must admit that I haven't heard much else from him.
4. James Horner. One of my long-time favorites. I've always liked his sound, and I love some of the themes he's composed (Braveheart, Titanic, all those epic and wonderful scores).
5. Alexandre Desplat. He first caught my attention in Girl With the Pearl Earring, so I followed his work. I bought The Painted Veil, and absolutely loved it (partially because I just love the movie so much). I then bought part of the Golden Compass, but was a bit disappointed, quite honestly...

HONORABLE MENTION
John Williams. A no-brainer. I must admit that, because I've grown up with such iconic themes such as Star Wars, Indiana Jones, Jaws and such, I probably don't appreciate Williams' great talent as much as I should. Williams' "Buckbeak's Flight" theme from Harry Potter 3 is probably one of my favorite themes. Ever.
(Only topped by James Newton Howard's central park, and Hans Zimmer's Love Theme from PotC3, as previously mentioned.)