Go into your Settings menu and select Accounts and sync. At the bottom of that, there's something that allows you to manage your various synced accounts. You can select the one you want to delete and press remove account. If this is the only gmail account on your phone, you may need to reset your phone or try adding your gmail account first before deleting theirs.
If you find that you can't add your gmail account through the gmail app, you can try signing in to your gmail account through your YouTube app and it will add.