What permissions are necessary for users to use and modify files?

Full control allows someone to do everything including changing permissions.

To be really granular, you need to click the advanced button.  Then you can set permissions that allow write, modify, but not delete, etc.

A better / easier option would be to redo your folder structure to more closely resemble the permissions people need.
http://technet.microsoft.com/en-us/library/cc782737(WS.10).aspx

NTFS can be fairly complicated or very easy and wide open.  You really need to read up on it though, because there's no easy solution that fits every environment.

http://searchwindowsserver.techtarget.com/generic/0,295582,sid68_gci1050093,00.html

This is a small 10 user domain so all domain users have the same permissions to the folder. Again what security permissions are necessary for staff to create, modify and save but not delete?