Music help?

What genre of music is Alexisonfire?

and what exactly is screamo?

Alexisonfire is considered to be post-hardcore/metalcore... in other words - just your average screamo band.
And screamo is just that... music that has all or mostly screaming vocals instead of clean vocals *regular singing*.

Alexisonfire could be considered as posthardcore, they have a lot of screamo vocals in their song, but always with some melodies in it. they're really really good indeed.
check out also these bands if you want to get into screamo: angercore, intohimo, your hero

I actually enjoy screamo.

I hate emo, but I like screamo. You don't even have to know the words to sing along. lol