I know no ones like the constraints of genres and there's much controversy over what these words are even supposed to mean, so I use the term "post-rock" with caution.
But, typically, "post-rock" songs are instrumental in nature, relatively lengthy, and elevate instruments (guitars usually) to generate differences in harmony, texture, and dynamics -- a lot more subtle than typical songs in which the vocals are the more critical force in shaping development. Vocals are usually absent, minimal, or more complementary than leading.
Some of the more well-known post-rock bands today:
Mogwai
Explosions in the Sky
Sigur Ros
The Appleseed Cast
This Will Destroy You
Godspeed You! Black Emperor
I'm personally a big fan of Explosions in the Sky's The Earth is Not a Cold Dead Place, and I feel like that's an album that brought a lot of kids into the genre. Appleseed Cast's two Low Level Owl albums are great, too, though not as dynamic as Explosions. Sigur Ros is probably a little different from their post-rock counterparts since the lead singer uses "Hopelandic" as a central force in the music. But their album Ágætis byrjun is still one of the most beautiful things I've ever heard. I can't get into Mogwai, though I do love their "Take Me Somewhere Nice" like crazy.
So any other fans? Did I make any glaring omissions? I know Explosions in the Sky and Godspeed You! Black Emperor is pretty popular these days.
(Topic title comes from someone's comment regarding Sigur Ros's sound.)