Grunge is a form of Alternative Rock. A very popular one, at that. So popular that it made alternative rock more popular, laying the foundations for all alternative bands to come, with the result being... kind of bad for alternative rock, actually, since the term "alternative" lost all meaning when alternative rock became the dominant form of rock music, and not so alternate anymore. Thanks a lot, grunge.
Ignoring that little Take That at grunge's expense... grunge itself is widely believed to have started in Seattle, and is sometimes referred to as "the Seattle sound" by nobody in particular, even though some bands weren't actually from there. Grunge was created by two groups - Malfunkshun and Green River and whoever tells you otherwise doesn't know what they're talking about - and started off underground, which was the philosophy of the music. As a backlash against Hair Metal, which was big at the time (the mid-'80s), grunge artists didn't care much for theatrics or their appearance, resulting in that trademark messy look characterized by flannel shirts.
The music itself could be described as a strange combination of sludge metal and Alternative Rock with even more influences from Hardcore Punk. The guitars have plenty of feedback (usually), and the lyrics are often pretty personal, influencing lots of more angsty genres such as Nu-metal and modern emo bands. The vocals range from Perishing Alt Rock Voice to Nose Yodeling, and all stops in between.
So, grunge started off underground, but of course something happened. What was this "something", you may ask?
Nirvana, a Seattle (by way of Aberdeen and Olympia) grunge band fronted by Kurt Cobain, unexpectedly made it big in 1991 with their album Nevermind, containing the famed hit single "Smells Like Teen Spirit", which, in typical "most famous song" fashion, was shunned by Cobain. With Nirvana's success, other grunge bands, such as Pearl Jam, Soundgarden and Alice in Chains, also made it pretty big. Against their will. Being big and famous went against grunge's philosophy, and many of the bands tried to avoid fame. Cobain himself was pretty depressed over this, as well as many other things, including his heroin addiction. This whole depression thing culminated in Cobain's shotgun suicide in 1994 (although some conspiracy theorists still maintain that he was murdered).
Then Nirvana disbanded, Soundgarden released their most famous song ("Black Hole Sun"), and grunge began to fade in the shadow of Britney Spears and a new generation of Millennial techno-geeks inspired by the Britpop revolution which was historically hostile to grunge. Another blow was dealt to grunge nearly a decade after Cobain's suicide when Layne Staley, lead singer of Alice in Chains, died of a heroin overdose -- on the approximate date of Cobain's death , which is cause for concern and even more paranoid Conspiracy Theories. Courtney Love did it!
So, there you have it. The strange story of grunge.
The "signature look" of grunge, much to the distress of fashion designers everywhere, was loose-fitting jeans and a t-shirt with an open flannel shirt over it. This had been a "burnout" look among poorer teens for years -- the members of grunge bands only wearing such clothes because it was literally the only warm clothing they could afford when they were struggling -- and was now promoted to "cool" kids.
Although grunge's time is over, the genre is still viewed as a landmark one due to its honesty and creative musicianship (especially compared to what it mutated into), and it still has a fanbase in certain corners, and the teens of the nineties are starting to look back upon it with nostalgia. Some of the bands have also had a significant influence on Doom Metal subgenres such as Stoner Metal and Sludge, which could be seen as a Spiritual Successor of sorts.
Not to be confused with Post-Grunge, which, beyond being based around downtuned guitar, often has very little in common.
Definitely not to be confused with Emo, unless you are referring to the early-90s Post-hardcore movement. Also do not confuse with Gunge (http://en.wikipedia.org/wiki/Gunge); the goopy stuff often dumped on contestant's heads during kid's game shows in the mid-90's.
- Both were discovered dead after some time had passed. Both were still listed as circa April 5th, however.