Now, I tend to pick my favorite music in terms of records, because when it's all said and done, does it really matter who made the record as much as the record itself matters? That being said, a lot of my favorite music is bands that made one or two albums and then disappeared into the abyss. It's unfair to say that these are my favorite bands, because they've only made forty or so minutes of enjoyable music. All of the guys I'm about to list have released several albums that I liked.
The Velvet Underground (as well as Lou Reed's and John Cale's respective solo careers)
Operation Ivy (& other Armstrong/Freeman bands)
Pixies
Fugazi (& other Ian MacKaye and Guy Picciotto bands)
Sonic Youth
Dinosaur Jr. (I have the utmost respect for their ability to reunite twenty years after they split and make another amazing album.)
Saetia, Off Minor, Hot Cross (three screamo bands sharing some members)
Weezer
Nirvana
NOFX
Elephant 6 Collective (huge group of guys who release a lot of stuff under different band names)
Tomas Kalnoky (member of Catch 22, Bandits of the Acoustic Revolution, and Streetlight Manifesto)
The White Stripes
Bomb the Music Industry!