Soundgarden, along with Nirvana, the Melvins, Alice in Chains and Pearl Jam, are all considered seminal grunge bands. Soundgarden, maybe more than these other bands, more directly influenced later rock music, at least in terms of sound. This may be due to the fact they were more directly influenced by bands like Led Zeppelin and Black Sabbath, who have themselves continued to influence music.
"Spoonman," above, is one of the most loved among critics. It is hard not to see why, since the song is an ode to someone who makes music using spoons, integrating the sound into a rock song. (And who writes a rock song about something that does not involve sex, drugs, angst or fame? Who would write a rock song about spoons? Soundgarden, that's who.)
Chris Cornell is probably still the most famous member of this band. After Soundgarden, he would go on to found Audioslave, and I am under the impression he does solo work. Soundgarden has also recently gone on a short reunion tour.