I briefly touched on the alternative genre in my write-up of Nirvana, but some of the terms I called up deserve further exploration, mainly to address how similar it is to classic rock. Again, much of this comes from my own knowledge; I'm not putting a lot of research into this and, as a result, my presented history may differ somewhat from reality. If you care, I suggest looking up some authoritative sources, as opposed to an amateur blog.
The easiest source to cite for alternative music is the post-punk movement of the early 80s. Though difficult to define, post-punk can be understood as any group with similarities to Joy Division (except for modern ripoffs like Interpol and She Wants Revenge; they both fit more comfortably into generic alt. rock) or Public Image Ltd, depending on the band's country of origin. Post-punk is noted as 'difficult' music; it owes a great debt to the Krautrock movement of the mid/late-70s and therefore uses a lot of synthesizers, odd time signatures, and abstract/absent melodic structures. There's a lot of focus placed on rhythm and on the twisting of melodies, making it the direct predecessor to drone, shoegaze, noise rock, and No Wave. Important figures include the aforementioned bands, along with the Velvet Underground, Lou Reed's solo career, Kraftwerk, The Stooges, Iggy Pop (especially The Idiot), David Bowie, certain Beatles' songs ("Revolution 9", "Helter Skelter", etc) (proving once again that they had their little hands in nearly everything), Brian Eno, and King Crimson (technically progressive rock).
For the alternative genre, post-punk is best understood through the lens of the seminal New York group, Sonic Youth. More likely to walk on their instruments then play them, Sonic Youth helped bring the genre more into the light of the day via a few surprise successes in the late-80s, particularly Daydream Nation. If you've ever read Pitchfork than you've no doubt heard enough fellatio for Thruston Moore and Kim Gordan, so I'm going to leave things at that. Other majors groups falling under the post-punk umbrella are Dinosaur Jr, Pavement (another favorite of Pitchfork), Nine Inch Nails, The Birthday Party, and Nick Cave & The Bad Seeds.
This is all very similar to the rise of punk in the late 70s; the post-punk/proto-alternative bands had some popularity and a lot of notoriety, but they were largely ignored within the pop scene. Sonic Youth were hardly a match for Madonna; after all, you can't really dance to "Teen Age Riot".
Also notable, but not fully under the banner of post-punk due to their more conventional song structures, were the Pixies. I've mentioned them before and I'm bound to mention them again; the Pixies are, in my mind, the modern equivalent of the Velvet Underground. While VU inspired punk and post-punk, the Pixies inspired the modern interpretation of alternative rock by way of Nirvana.
I've covered this before so I'll stick with a summary: Nirvana basically took the basic formula used by the Pixies (soft-loud-soft song structures) and combined it with melody, thus defining alternative rock for the early-90s.
I'm avoiding the use of 'grunge' because it's a terribly misapplied. Grunge only describes one Nirvana album (Bleach) and refers to a lo-fi sound best equatable to garage rock. Dinosaur Jr was grunge; The Meat Puppets were grunge; PJ Harvey's first two albums were grunge; Nirvana was only grunge when Sub-Pop forced them to be.
Which brings me to my next point; alternative rock is horribly bloated. The same genre contains any number of groups that sound nothing like one another; the link between Soundgarden and Smashing Pumpkins is tenuous, at best. One owes a debt to Led Zeppelin and other first-wave metal groups, while the other pays homage to baroque pop with post-punk sensibilities. Do you realize Bjork is considered alternative? How does she sound anything like VU (aside from the Nico comparison)? It's all nonsense, just like classic rock.
The 'grunge' of the early-90s gradually gave way to the much maligned adult alternative of John Mayer and the Dave Matthews Band, as well as the even more bizarrely named post-grunge genre, which was basically alt. rock with the piss taken out of it (Matchbox 20, Nickelback, Goo Goo Dolls). Post-grunge is best compared to New Wave in terms of genre relations; a follow-up movement specially calculated to be mainstream. I'd call it No Wave, but that one's been coined already. How about fuzz pop? It sure as hell isn't 'alternative', considering it was manufactured for the radio.
That last bit, by the by, is the reason I think alternative rock imploded in on itself; the artists involved didn't want to be popular. They were alternative after all, it was the same pitfall punk hit back in the late 70s, and it was solved by radio in the exact same way.
I should also bring up the fact that alternative rock technically includes nearly every genre that emerged in the 90s, including rap rock and funk rock. This is ridiculous for reasons I shouldn't even have to address.