For now, 360-degree YouTube videos are only fully supported in Chrome for desktop and the YouTube app for Android. In Chrome you can use your mouse to drag the point of view, while in the Android app you can move your device around or manually change the point of view. If you watch these videos in other browsers or apps, you can see the entire panoramic images.
Here's a playlist with 360-degree videos:
If you want to upload your own photosphere-like panoramic videos, there's a help center article that offers more information. YouTube supports 5 cameras: Bublcam, Giroptic 360cam, IC Real Tech Allie, Kodak SP360, Ricoh Theta. For now, you need to run a script to insert metadata, but YouTube will try to make it easier to upload videos in the future.