Once you select a username and click Next, go back to your Apple Device and enter in either your email or username and your password.
Some apps, like the YouTube iOS app, can use an application specific password that's independent of your normal password. If you're having issues signing in with your normal Google account password, I'd recommend to enable 2-step authentication so you can generate a new password. Here's a Help Center article that walks you through the steps:
When you enter in the name of the application at the bottom of accounts.google.com, I'd recommend a name like YouTube on iPhone.
Note: Once you have 2-step authentication turned on, you'll always need to use your application specific password. If you sign out of your Apple Device and need a new password, you can always re-generate one by visiting accounts.google.com.
If you're still having issues with signing into the iOS app on any Apple device, I highly recommend going to m.youtube.com in Safari.
For any additional issues with your Apple device, you may want to contact Apple at http://www.apple.com/support since Apple currently owns and develops the YouTube app on all Apple devices.