If you are watching a Tim Burton movie, you are listening to a Danny Elfman soundtrack. The two are inseparable. Batman and Batman Returns, Edward Scissorhands, Alice in Wonderland, Beetlejuice, Charlie and the Chocolate Factory, Sleepy Hollow, Mars Attacks! He also frequently collaborated with Sam Raimi, such as in the first two Spider-man movies, though they grew apart after the second. His music often incorporates vocals in a very instrumental way, not with lyrics, but with oohs and aahs. Beautiful stuff.
James Horner has some pretty tunes, often stirring feelings of yearning for me. Think of Don Bluth cartoons like An American Tail, Land Before Time, etc. Also movies like Willow, Searching for Bobby Fischer, Field of Dreams, The Legend of Zorro, Star Trek II: The Wrath of Khan. Most famously, he composed the soundtrack for Titanic, including the once enjoyable, but so over-played that I can't stand it anymore, My Heart Will Go on.
Vangelis is more of an electronic composer, writing soundtracks like 1492 or the very famous Chariots of Fire (think of all those movies where you see someone running in slow motion, hear the music playing, and laugh at the reference to those guys in short shorts running along a beach in the original film).
James Newton Howard has created some good stuff, often exciting in my mind. He has worked a lot with M. Night Shyamalan, so think of The Sixth Sense, Signs, etc. He also composed for The Fugitive. But his best move was his collaboration with Hans Zimmer on Batman Begins and The Dark Knight, focusing on the more dramatic parts of the movie and utilizing more electronic music while Zimmer used more orchestration and focused on action.
Bernard Herrmann (oops, two r's) became familiar to me through my love of Alfred Hitchcock movies, and he composed for films such as Psycho (the famous screeching during the stabbing), North by Northwest, The Man Who Knew Too Much (even directing the orchestra during the assassination attempt), and my personal favorite, Vertigo. You also might have heard him on The Day the Earth Stood Still. I believe I read that Danny Elfman took inspiration from him, and it is pretty apparent. This man was great.