One of these is the "set user ID" bit. If this bit is set on an executable program, then whenever the program is run (by someone otherwise authorized to do so) the effective user ID of the process running it is changed to that of the file's owner. Thus, the program is able to access files according to the rights of the owner of the program, rather than the actual user. (The user ID reverts to its actual value when the program terminates.)
Note: In a long directory listing (ls -l), this bit shows up as an "s" where the "x" for owner execute permission would appear.