There are two great tools that help teachers build lessons around a YouTube video. Perhaps the most well known is TED-ED - yes, you guessed it--the sister site to those great TED Talks. What makes TED-ED special is that anyone (yes, that means the likes of you and me!) can create their own video-based lessons. With TED-ED, you can also 'flip' other lessons, that is, you can take a lesson someone else has designed and customize it.
Here is a short video that illustrates the features of TED-ED:
While TED-ED has the ability to monitor the progress of people taking the lesson, it isn't that well set up for school use. Users need to have a TED-ED account, and there is no way to segregate users taking one lesson.
That is where EDpuzzle comes in - they have taken the same TED-ED concept to 'build a lesson around a YouTube video' and wrapped it in an interface that lets teachers create individual classes and deliver their own video lessons to their students. But, what is more, EDpuzzle lets a teacher create a project in which students have to find their own videos and build their own lessons. The interface is fairly straightforward and student projects uploaded are private between the teacher and the student. At the moment, it isn't possible to upload videos there, so if you want to create your own video then you have to publish it on a media server somewhere, like Vimeo or YouTube. However, you can make your video 'unlisted', so for all intents and purposes it is hidden.
If you want more ideas on how to use this with students, the folks at EDPUZZLE have provided some materials you can download from this DROPBOX folder. Enjoy!