Following the removal of default YouTube app from iOS 6, Google has released its own YouTube app for the iPhone and iPod touch.
Watch the world's videos and keep up with your favorite YouTube channels with the official YouTube app for iOS. Sign in to access your subscriptions, playlists, uploads and more.
Features: ✓ Enjoy YouTube's vast video catalog, including official music videos ✓ Find videos and channels more easily with voice search and query autocomplete ✓ Subscribe to channels and instantly access your subscriptions with the channel guide UI ✓ Read comments, browse related videos, enable subtitles and more - all while watching ✓ Easy video sharing to Google+, E-mail, Facebook and Twitter
Google says a version of the app for iPad is also in the works.
You can download YouTube from the App Store for free.
I found it in the US app store but the second time by typing YouTube exactly and then scroll down to the 7th or so item. You'll find it keep searching.
If you go to the article about the release of the YouTube app on the cult of Mac site they have a direct link to the app that opens up the AppStore. That's how I got it. Couldn't find just by searching in the store. Hope this helps.