Click the menu button and select "My Channel." A "Sign Out" button is in the upper right corner. Once you have signed out, you may sign in using a different user ID, or add an additional user ID.
You will remain signed-out until you sign back in (go to My Channel to sign back in). You won't be automatically signed-in if you close and restart the app.
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When you're on YouTube, in the top right corner you will see a box that has the name of your account on it, click that and at the bottom of the list of options it should say "Sign out". Click it, and you've logged out.
This will only work if you have a account on YouTube, if you don't then you don't need to log out.
Hope this helps!
The button to log off of YouTube is a little hidden. However, all you have to do is click on your channel name at the top right hand of any YouTube page and when it drops down, you can click the "sign out" button.