How do I disable the beep in OSX Leopard when I perform an invalid action?

Unchecking the "User interface sounds" doesn't disable all sound. But sliding the "Alert Volume" option all the way to the left (minimum), together with unchecking the other options does the trick.

Apple, System Preferences, Sounds.

The OP mentions using headphones while using his/her computer, implying that he/she would like to disable the terminal bell but allow outher sound output to continue. Adjusting the sound settings in the system preference panel won't solve this problem.

Readline rings the bell at the command prompt in the terminal whenever the tab autocompletion can't find any matches, or finds too many matches. To disable just this behavior, check your home directory for the presence of a .inputrc. If doesn't exist, add it, then in either case add this line:

set bell-style off

Source this file

$ . .inputrc

or quit the terminal program and restart.

To completely disable sounds in the terminal, edit the apps preferences: Terminal menu, Preferences, Settings tab, then Advanced tab on that screen. There are two check boxes: audible bell and visual bell. Adjust to your tastes accordingly.