Last updated 20 April 2020: Clarified how lazy-loaded videos can be tracked with this trigger. Let's cut straight to the chase. Google Tag Manager has just released the YouTube Video trigger, which gives you native support for YouTube video tracking. And it's great! Even though we've been more than satisfied with the excellent tracking scripts provided by e.g. Cardinal Path and Bounteous (with a small modification from yours truly), this is a no-brainer for native support in Google Tag Manager.
Tracking YouTube videos in Google Tag Manager is one of the more useful things you can do in terms of tracking. YouTube has a wonderful API that you can tap into and convert the detected events into dataLayer messages. There are some really good solutions out there for tracking YouTube videos in GTM: The wonderful solution by Cardinal Path. An even more thorough treatment by Bounteous.