Apple - 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 buying/downloading it yourself. If that's the case...

  1. Delete the application
    • open Finder
    • select Applications folder in left pane
    • drag apps tied to the old account to trash
  2. Ensure that the old user is signed out of the Mac App Store
  3. Sign in under an account you control
  4. Purchase/download to your heart's content