I did this one at Dave's suggestion.
The name "post-punk" was first used by Greil Marcus in a July 1980 issues of Rolling Stone when referring to bands like The Raincoats and Gang of Four.
This genre is rooted in punk rock but progressed into something more complex as time went by. It is for all intents and purposes the precursor to the alternative rock genre (and you know how much i love that genre). It also led to genres such as gothic rock and industrial rock.
The genre got its start after 1977 after many bands who wanted the capture the spirit and rawness of punk rock but wanted to experiment a bit more and explore. So post-punk was formed which is basically a more arty version of punk rock.
Bands such as the Psychedelic Furs, Devo, Siouxsie & the Banshees, Joy Division, Killing Joke, The Cure, etc...kind of fell along the lines of post punk in its early forms.
It has seen a bit of a revival lately in the form of bands such as The Killers, The Strokes, and the Libertines.
Here is a list of some other fairly well-known post-punk bands:
Adam and the Ants
Big Black hey wait! they were math rock last time!!
Dead or Alive
Echo & the Bunnymen
The Human League
Public Image Ltd
Sisters of Mercy
U2 I'm not really sure if I would lump them in this genre but I can see why someone else would.
Here is a list of my source material (also at Dave's suggestion):
Wiki article on Post-punk.
The Allmusic article on post punk.
Wiki list of post punk bands.