To sign into this app on your mobile device : iPod touch, iPad or iPhone
Get your "Application Specific Password"
from your Google Account ( http://accounts.google.com ).
hopefully this video : http://youtu.be/W61rO8SdqaY
or the attached pics below clarifies this :
Step 1 - Click on Authorizing applications and sites
Step 2 - Name it then Click on Generate Passwords

Step 3 - When generated the Application Specific Password
is displayed inside the Yellow Frame ( as shown below )

Finally go to the Youtube App on your iPhone, iPad or iPod touch
USERNAME : youtube username ( please note this username is associated with you Google Account )
PASSWORD : Application Specific Password
I tried to simplfy the process above from m.youtube.com :
I managed to sign in to youtube through the app. After approx. 25 tries
I really hope this helps.
Regards