First, check your version of iOS by going to Settings>General>Software update. If there is an update, you should update. By default, iOS 6 will not require a password to update apps. If it still requires a password, go to Settings>General>Restrictions. It will ask you for a restrictions pass code. Enter that (it could be different from your lock screen pass code), then scroll down to "Require password." Tap that, then tap "15 minutes." Switch back to the App Store, and you should be able to update.
Also, if this is your apple ID, you should reset the password.