Eh, you're picking out some
very random genres here to be honest, haha. Rock has thousands and thousands of subgenres (although a lot of them are quite ridiculous) - so singling out just emo, punk, progressive and post-rock is quite odd, lol (since there isn't really an obvious connection between them). I'll start off with emo and punk, which is the easiest in this situation. Punk, being an offspring of rock & roll, is actually a genre that has many subgenres, emo being one of them. So what is punk? Depends on what kind of punk we're talking about. It's actually too big a genre to summarize it shortly, although this quote from allmusic.com may give you an idea:
Read more on
All Music Guide.
As for emo, it's a term that's being abused heavily these days. Bands like Green Day, Simple Plan, etc., are being reffered to as emo while they are in fact pop-punk. Emo originated in the 80s, with some of the important bands that kickstarted the genre being Rites Of Spring and Embrace (members of Embrace later went on to form Fugazi, one of the best-known emo bands in the world). To contrary belief, emo is not whiney music made by guys with tight t-shirts who like to cry a lot and write poems. It originally was a quite punk-rockish genre that set itself apart from punk by relatively melodic guitars and such. Over the years it evolved into something much louder and chaotic (key bands: Heroin, Mohinder) and eventually turned out to sound much like hardcore, but with more melodic guitars and vocals - sounding as such more emotional indeed (examples: Envy, Thursday).
Here is a *great* site on emo, you can find out what it really is here (and it's not Green Day, and it's definitely not Coldplay!).
As for progressive, I'll pass on that one because I couldn't care less about it, haha. I'm sure there's people here who know more about it.
Post-rock started out around the mid-nineties and composition- and instrumentwise it's often quite far from regular rock. Ten-minute songs are quite usual in this genre and yatta yatta, well, haha, you should just hear it (key bands: Explosions In The Sky, Mono, Godspeed You Black Emperor!).
Want more info, just ask. Exhausted now so all this seems chaotic and messy, sorry about that. You can always check out allmusic.com for some good info.
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As for nu-metal, it's a genre that came into existence somewhere in the late nineties, often combining rap rhythms, metal riffs, and a funky overall feeling (although this isn't a general rule). Key bands: Korn, Linkin Park, Limp Bizkit, and any other lame band like this you can imagine (from the top of my head: Sevendust, Taproot, Coal Chamber, Soulfly). To clarify: none of Ayu's music is nu-metal, although you can sometimes hear influences by this genre. However, Ayu's music nearly never exceeds the pop genre.
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Just check allmusic.com, they have the most reliable genre definitions, leave for a few.