From the boot menu, select recovery mode, which is usually the second boot option.
After you select recovery mode and wait for all the boot-up processes to finish, you'll be presented with a few options. In this case, you want the Drop to root shell prompt option so press the Down arrow to get to that option, and then press Enter to select it.
The root account is the ultimate administrator and can do anything to the Ubuntu installation (including erase it), so please be careful with what commands you enter in the root terminal.
To reset the password, type
where username is the username which you want to reset. You'll then be prompted for a new password. When you type the password you will get no visual response acknowledging your typing. Your password is still being accepted. Just type the password and hit Enter when you're done. You'll be prompted to retype the password. Do so and hit Enter again.
Now the password should be reset. Type
to return to the recovery menu.
After you get back to the recovery menu, select resume normal boot, and use Ubuntu as you normally would—only this time, you actually know the password!
If you are in a weird situation in which your Grub boot menu timeout is 0 seconds, so you cannot select recovery mode or edit any kernel lines, you can use a live CD to reset your user password: