If you download a map or mod and it has a bunch of files in the archive this is the general rule:
.unr files go in /maps
.int, .u, .ini and .dll files go in /system
.utx files go in /textures
.uax files go in /sounds
.umx files go in /music
readmes, instructions and such can go in /help (optional)

UMOD files were supposed to be the self-installers. These days there are a few ways to use them, but my chosen method is to use an external installer program.

UModBrowser_Setup.exe
http://wiki.beyondunreal.com/UMod_Browser

This tool will give you access to the files inside of a UMOD. You can set the path to your current local installation of Unreal Tournament and it will install whatever you have loaded.

Unreal Archive
https://unrealarchive.org/
This is a preservation effort for the old UT99 community generated content. You can get maps, map packs, mutators, models, skins, voice packs, patches, and guides. Excellent site.

Thanks to 'Showtime, Synergy!' for posting this information in the forums.

6/30/2013 - Guide Posted
3/12/2018 - Added umod installer info. Changed the incorrectly noted 'GOTY files' to 'UMOD files'
6/21/2020 - I have been alerted that the links are broken. My hands are full at the moment, but as soon as I can, I'll go find some map sources to replace them.
6/23/2020 - Updated the links section with tested, working websites. I was able to download usable files from these sites as of today. Big thanks to user 4exan for pointing out the old ones were dead.
1/21/2021 - Fixed the link to a working umod manager. Made sure the map sites were still up and those links still work.
3/24/2022 - Made sure the links still work.
9/27/2022 - Made sure the links still work.
11/6/2024 - UT99 maps, (the site) has apparently closed. Removed it from the links section.