In recent times with the introduction of iOS 6.0+ YouTube and music lovers had to give up the old native YouTube app for a slightly below par standard application. One of the great features the old app used to have was the ability to listen to a music video with the phone locked or the application minimised.
My house mate recently recommended an app called TubeBox which I bought from the Apple Store for £1.49 (I believe there is also a free version available). This fantastic app not only lets you play music videos in the background when the phone is locked or you are on another application but you can download and save videos to your iPhone to playback when you haven't got a internet connection.
So just to give a little demo and let you see it in action:
The above left photo is the metro interface-like main menu where you can access your saved videos, which can be ordered into different sections. Selecting the middle option at the bottom allows you to browse all of YouTube or if you sign into your YouTube account you can view your personal playlists, favourites and subscriptions (Right photo).
Its as simple as browsing a list of videos and selecting which one or more you want to download. Its worth noting that TubeBox by default will download by all videos in 720p ( or the highest available quality ).
PS if your big on your music check out my friends channel :
http://www.youtube.com/user/undergroundliquid