Musically, the band is often associated with the noise rock scene, and early releases were somewhat experimental in nature. Not unlike earlier experimental bands (like the Velvet Underground) this resulted in critical acclaim but only modest record sales. Over time, Sonic Youth's sound evolved and songs incorporated more traditional rock structures and sounds, though they maintained a bit of an experimental feel. This music was loved by alternative musicians and helped lay the groundwork for the 1990's. [As an aside, if this sounds like Sonic Youth took a similar path to Hüsker Dü, well ... they did.]
"Teenage Riot" was the song that blew the lid off Sonic Youth in the emerging college rock scene. Although the song is still edgy, it also has a more traditional structure and sound, which found a new and larger audience for the band. While Sonic Youth never became mainstream stars, they did become heroes in the college rock community, and provided an inspiration to countless 1990's bands -- everyone from Nirvana to Nine Inch Nails.
Sonic Youth remained together and relatively vibrant until Kim Gordon and Thurston Moore divorced in late 2011 (after being married for 27 years). While no official announcement has been made, it appears that this marked the end of the band.