Alright, I'm fairly new at all of this mapping stuff, and I have a few quick questions.
1) When I have downloaded a map of the website, where do I save it and how do I run it?
2) Is there any extra software needed to be downloaded in order to run the maps on the site? (I have the Hammer Editor)
3) If I want to co-operatively play on a downloaded map with a friend, how do I run it?

Any help, any help at all would be greatly appreciated.

1. You'll want to save it to F:\Program Files (x86)\Steam\steamapps\common\portal 2\portal2\maps. Or wherever you have Steam installed.

2.Nope

1. You'll need to invite your friend to a coop game, then from inside the hub open the console with the "`" key (located right to the left of 1, and type changlevel nameofmaphere.