According to Wikipedia Post-rock is a form of experimental rock characterized by use of rock instruments primarily to explore textures and timbre rather than traditional song structure, chords or riffs. Post-rock artists typically unify rock instrumentation with electronics, and are often instrumental.
However it should be noted that post rock is quite experimental music and it has lot of connections of indie music, at some point it blends in.
Below you can see so called periodic table of post rock, I do quite few of groups listed here, but still there is lot more to explore.
I would suggest everyone try listening to post rock, I would like to suggest starting with some instrumental groups like God is and Astronaut or If These Trees Could Talk.
If you are fan of bit more heavy music you should try Russian Circles or Pelican an also great instrumental bands.
One of most popular post rock bands is Joy Division, nowadays we can see lot of people wearing their shirts, while they have no idea about music in general. You may have seen shirt pattern like this :
It is from Joy Division Album called unknown pleasures. While some may argue, that JD are not post rock, but are post punk indeed, line between both is quite thin and most of these 'post' bands diffuse together quite a bit.
Regarding JD you may have heard this song before :
This one is my personal favorite and kind of started my obsession with Post rock bands in first place.
If you are looking for something new and are fan of rock, you should try post rock, you may be surprised !
Some of post rock and similar to it bands I suggest to check out are - Motorama, Explosions in the Sky, Brandenburg, Mono, Manicure, Soviet Soviet, Acid ghost, The Blackmail, In a lonely place, Hammock, Radiohead, This Will Destroy You...