App Store asks me for a password, but won't let me change the Apple ID
It won't let you change it because the apps are tied to the account that purchased them.  See here for the legal details.

Basically, only the original purchaser can update.  If you'd like to be able to keep up-to-date with an application, you must have it tied to your account...by buying/downloading it yourself. If that's the case...


Delete the application


open Finder
select Applications folder in left pane
drag apps tied to the old account to trash

Ensure that the old user is signed out of the Mac App Store
Sign in under an account you control
Purchase/download to your heart's content