YouTube videos on ATV?

From the YouTube option in the main menu. You should setup a YouTube account in your computer and then use the same account in the YouTube section of your ATV, so you can access your favourites or your browsing history.

You can also airplay a youtube video from your iPhone or iPad using the YouTube app on such devices.

You can search for most viewed or specific ones. I have ATV3 but should work the same on all ATV versions.