I can tell most of the suggestions so far have been by Americans, lots of Rock and Grunge.
As a kid born in 1987 in the Netherlands the 90's was all about more electronic music. Things like House (
or EuroHouse as you Americans call it), (happy) Hardcore and Gabber. It was pretty common to even hear it in the supermarket. We still got a lot of rap, RnB and Rock of course but it was a pretty eclectic mix.
Daft Punk - Around the world (1997)
Jamiroquai - Virtual Insanity (1996)
Party Animals - Have You Ever Been Mellow (1996)
Captain Jack - Captain Jack (1995)
Mr Oizo - Flat beat (1999)
KRS-One - Sound of da Police (1993)
Stardust - Music Sounds Better With You (1998)
That's also some early Daft Punk.