Post-hardcore is a label that, from what I can tell, is basically just an offshoot of hardcore punk. While hardcore punk was defined by music that had a very fast tempo and heavy bass, post-hardcore went outside this definition with more complex arrangements, going beyond the typical power chord arrangement that is definitive of most punk rock bands. In addition to moving beyond the complex arrangements, the music typically mixes elements of other styles of music such as jazz and soul.
Personally, I think a post-hardcore punk band is basically what I would call a bunch of guys from a formerly punk rock band that learned how to play their instruments. LoL!
The Washington DC hardcore scene was highly influential in the origins of the post-hardcore movement, which was led, in part, by a number of the bands that released their music through the DIY (Do It Yourself) label, Dischord Records.
Some of the bands included in a list of post-hardcore bands would be:
Girls Against Boys
Hot Water Music
June of 44
Rites of Spring
The Nation of Ulysses
The Jesus Lizard
If I had to choose a list of favorite post-hardcore acts, I would go with Fugazi, Emery, Big Black and Helmet.
In many ways, post-hardcore sounds a lot like indie rock. The main difference between, in my opinion, by who the particular bands would quote as influences. While a post-hardcore band would, of course, list mostly hardcore-punk and other punk rock bands, an indie rock band would list its influences in mostly pop bands and bands of other types of rock music.
You can read a lot more about the genre itself, as well as a pretty decent history of the genre at the Wiki article here.