How do I download an app to an iPad that asks for another user's password?

Solution 1:

The app was downloaded / bought using your ex-girlfriend Apple ID; delete and re download the app in question.

Solution 2:

Two things:

1. Open the settings app and App Store - sign out of the account.
2. Then make note of all apps that need an update. You'll have to determine by which account pops up, which account bought the app in the first place.

Delete all the apps that are assigned to the account you wish to remove from the device. Sign in only with your account going forward and delete all the apps (either waiting for them to update or clearing them out more pro-actively).