In no particular order..
Fishbone: These dudes epitomized that hardcore punk-funk fusion with a hint of ska.. They also typified that existential "LA" approach to everything..
Eagles: Perhaps these guys tried a bit to hard, but their impact on mainstream American music is undeniable.
Rage Against the Machine: Was there ever a more pissed off band on the radio?
Descendents: Punk could be fun, much like Los Angeles, this band is really a precursor to the new-wave --> grunge/alternative eras where music sounded loud and obnoxious like punk but was not as serious..
Black Flag: The opposite side of the Descendents coin, one of the seminal punk bands.
tool: Metal can be smart? Philosophical? Talented? People from LA can also be all these things? It was a surprise to everybody..
The Beach Boys: These guys weren't just having fun, I think we all accept Pet Sounds as one of the most experimental and influential records of all time..
L7: While most "riot grrl" bands couldn't crack their way into the mainstream, L7 handled business.
Red Hot Chilli Peppers: It was said on another thread this was a "singles" band rather than a band with great records. Agreed completely, but their singles were part of what defined the late 80s and early 1990s..
Ice-T: A lot of other rappers kind of stole his thunder and spotlight, but Ice-T essentially invented "gangsta rap" and this should not be minimized..
Sublime: Totally misunderstood, but inspired the entire "ska-ish" era of the mid-to-late 1990s which was insanely popular..
System of A Down: The self-titled record IS LA sound.. fierce, exotic, some what hip-hop, pissed off..
The Byrds: No explanation necessary..
Motley Crue: Yeah these guys suck, but they also were a lot of fun when the 1980s America was all about partying , and partying to the point of excess is LA's favorite cultural past time
Rilo Kiley: A lot of people really enjoyed this band, and they were part of the pioneers of what has become the current LA scene.. female vocals..keys.. charming guitar work.. Essentially EVERY LA band today sounds like this (albeit blended with shoegaze)
Suicidal Tendencies: Most punk bands cite these dudes as an influence, must be noted.
WAR: Totally underrated, totally Los Angeles (multicultural, politically conscious, fun)
Weezer: Is it safe to say that these guys invented emo?
The Doors: No explanation necessary..
Van Halen: This was one of the biggest bands of the 80s, and was the first "metal/rock" band to have a female audience, essentially integrated music from the previous gender gap of the 1970s, before Van Halen women simply didn't attend rock/metal shows, after? They make up at least 1/3 of most crowds if not the balanced 50-50..
(some of these bands I dig, some of these are absolute shit, but ALL them contributed in some significant way to the fabric of American musical culture)