1. Make sure you have both Stuffit Expander and Disk Copy on your desktop. If they're not there, go to the folder Applications=>Utilities , find these programs, select the icons, press command(apple) M and drag the aliases to the desktop.

They should both be installed with the system, but if for some reason you don't have them , here are the links. Make sure you update to the latest version of Stuffit Expander (6.5):

Stuffit Expander 6.5.1

Disk Copy (OS7.x to OS9.x)

2. Drag the downloaded files directly over the Stuffit or Disk Copy icons.

.sit, .bin, .hqx, .zip(PC) to Stuffit (will decompress the file)
.img (OS7-OS9) .dmg(OSX) to Disk-Copy (will mount the disk image)

Double clicking commpressed files or disk images will sometimes open Limewire or cause other unwanted actions.

3. There are also a number of .sit / .dmg files that simply refuse to mount or decompress. For example, if someone has partially downloaded a such a file and moved it to their shared folder, it won't open. You can check for cases like this by searching for files of the same name in your client and checking that they're the same length. If the file you've downloaded is smaller and doesn't open, then it's probably incomplete.