Great thread idea!
Here's my list. All of these have great music and are also great movies:
Amadeus (Classical, Mozart) Others have posted links on this so I won't repost.
The Chorus (Les Choristes) (Boy's choir, French. A great, touching movie!)
The Sound of Music (Absolute classic movie!)
Schindler's List
Ray
Godfather
Doctor Zhivago (Absolute classic!)
The Good, the Bad and the Ugly
Bohemian Rhapsody (Classic rock, Queen)
Les Miserables (Broadway)
Phantom of the Opera (Broadway)
Dirty Dancing (70's dance)
The Piano
Mr Holland's Opus
Saturday Night Fever (70's dance)
Drumline (College sports band)
Immortal Beloved (Classical, Beethoven)
Amelie
Once
A couple of movies that I haven't seen, but know they have great music:
Barry Lyndon (Classical)
La Vie En Rose (Chanson, Edith Piaf)