Logged on to someone elses IPad
You (or your friend) will need to delete all the apps you downloaded to your friend's iPad. The iPad is likely attempting to download updates to the apps you downloaded with your Apple ID, which your friend cannot do with her AppleID.

There's no great harm in leaving your apps on the device other than they won't back up for your friend and the nagging password prompt.

If your friend wants to clear out all of your apps in one step, they can back up to iTunes, transferring all purchases and then restoring the device. No apps locked to your password will make the jump to the backup and your friend will be left with the device cleaned of all of your apps.