The aptly named YouTube Picture in Picture extension for Chrome makes YouTube on your desktop or laptop feel more like YouTube's mobile apps where you can swipe down on a video to watch it off to the side so you can continue to browse YouTube. The extension is marked as being in beta, but I found it to be stable and dead simple to use. The developer is right, however, when he says that playlists don't work correctly with the extension.
After adding YouTube Picture in Picture to Chrome, you'll find the extension does its work automatically. Unlike with YouTube's mobile apps where you swipe down on a video that's playing to move it to the corner so you can browse YouTube's other offerings while your video plays, YouTube Picture in Picture requires no special gesture on your part. When a video is playing, you need only to click the YouTube button at the top of the page to dispatch the video currently playing to the lower-right corner of Chrome while you browse YouTube's homepage. You can also click the menu button in the upper-left corner and choose a playlist, one of your subscriptions, or any other YouTube page from the menu panel to open it and enable the extension's picture-in-picture window.
From the extension's picture-in-picture window in the lower-right corner, you have access to YouTube's usual playback controls along the bottom edge as well as a few controls at the top of the video player. At the top, the extension provides three buttons to move the video back to its original position within YouTube, toggle between large and small sizes for the picture-in-picture window, and close the picture-in-picture window.
The extension hiccupped every now and again when I moved a video in and out of the picture-in-picture player, but it was usually able to keep both audio and video playing smoothly through the transition. One thing the extension doesn't do is play well with playlists. You can't start a playlist and have videos autoplay one after the other in the picture-in-picture window.
Lastly, the extension adds a line to the right-click menu. Right-click on a thumbnail and choose Play Picture-in-Picture to start a video directly in the picture-in-picture window and bypassing YouTube's usual video player.
In related news, you can use Sideplayer for Chrome to watch YouTube videos in an overlay. On an iPad, learn how to use iOS 9's picture-in-picture feature with YouTube. And finally, here are three websites to watch YouTube together when you are apart.