In this example, I will show you how to display a different menu for users based on their login status for the WordPress TwentyTen theme. This process can be adapted to any theme (and using the same principles, could be adapted to use anywhere on your WP site).
Note: There is an alternate (and easier) method of doing this posted here. That post displays the method currently used on this site.
There is now a Logged In Menus Extension that makes the entire process as easy as installing a plugin.