Who owns the copyrights to the modules?
The composers do - unless the Public Domain license attribution has been specifically attributed in which case you are free to do with the module whatever you like.
Can I use a module of modarchive.org in my game/application/etc...?
If the module in question has license deeds attributed to it, please refer to those.
Otherwise, the only way to be genuinely fair to the artist who's music you want to use is to contact them and get their permission. We can't grant permissions on their behalf. Period.
Can I redistribute the module in its original unmodified form?
Yes. All uploads to the site are governed by an upload agreement of which one of the terms is the right to redistribute the original unmodified module file. This does not cover inclusion in a packed/bundled application or game. See previous item.
The contact information isn't in the module, what can I do to contact the artist?
You can try using the modarchive.org forums, specifically the Wanted: forum. Search the Internet, do some homework. If you fail to find the necessary information then you are pretty much out of luck.
I can't get in contact with an artist who's music I want to use, what now?
It's up to you how you proceed. We will not condone illegal use of someone's works, but if you still wish to continue to use that particular piece of music in your project, then you take on those risks yourself.
If your project is small and free (non commercial) then it's wise to give clear and concise due credit to the artist in your production. If your production is for commercial purposes then you are technically left with no other choice than to find a different module where the artist has attributed a compatible license, or is contactable - or go out on a limb and ask to have someone compose a custom module for your project.
Where can I find the license attribution information on a module?
Look on a module's information page, there is a section dedicated to the License Attribution information.