It may not be immediately obvious how to change a password in Ubuntu and have been asked a couple times on how to do so, so I thought I'd share these simple instructions...

To change your password in Ubuntu, go to System > Preferences > About Me and click on "Change Password..."

This is also where you can change your display picture that is used on a multi-user login screen and when your computer becomes password protected.

EDIT: As others have pointed out, if you're using Ubuntu 10.04 (Lucid Lynx) - and still have the "Me Menu" on your panel - you can click your name then your avatar to pop up the same dialogue as System > Preferences > About Me. If you removed the "Me Menu" out of the panel like I did, you can re-add it by adding the "Indicator Session Applet". Why the name difference? I'm not sure.