(Rest in Peace: Layne Staley, Kurt Cobain, Mike Starr, and Andrew Wood. You were truly the inspiration of a generation along with: Jerry Cantrell, Chris Cornell, Eddie Vetter, and David Grohl who are still rocking.)
It originated with the jangle pop guitar sounds of R.E.M. and the loud/soft dynamic of the Pixies and gained popularity when Nirvana brought it into the mainstream with their album Nevermind. Ironically, it became probably the most popular music of the time period, which was a paradox of sorts in itself -- although purposely anti-mainstream and "against the norm", it became commonplace for people to conform to non-conformity.
Today, the term is often thrown around by people who don't really understand what it means and think that it can be applied to any band of their choosing. This is not true -- for a band to be truly Alternative, they have to follow the Alternative mindset; that is, they must either perform in a style of Alternative Rock already established (see Grunge or Britpop) or be avant-garde and completely throw what it considered usual or average out the window.
The name is often wrongly associated with "Modern Rock". This is wrong because whereas Alternative Rock is a genre and follows a specific style, Modern Rock is a radio format and simply refers recent bands.
Note: discrepancies with categorizing bands as "Alternative Rock" come from the fact that most Alternative bands fit easily into an Alternative Rock subgenre (Nirvana - Grunge, Modest Mouse - Indie) and can be classified as either the subgenre or "Alternative Rock" in general; if a band covers more than one style instead of sticking to one in particular, they are usually just called "Alternative Rock".
The label “Alternative” often applies to new music/bands before they become more widely popular due to creative people whose music does not fit into current genres.
Fred: Mostly alternative rock stuff.
Mike: Oh, cool. So you like Local H and Tool too?
Fred: Actually, I was talking about Puddle of Mudd.
Mike: ... Fuck you.
As punk was defined more by an attitude and an ethos than a particular style of music, so is alternative rock. That ethos is bands joining together all over America to make bland, passionless insincere rock music with lyrics about depression and breakups to cash in on teenage angst and stupidity.