I have been seeing this a lot lately and I am unable to find any information on it.
Is there a name for it?
Which browsers support this?
It's the same as an absolute URL except that the schema will be the one used for the base URL. Thus, if you say //mysite.com/images/img.png
on an https page, the image will be loaded with https
and on an "ordinary" http page it will be loaded using the http
schema (and, respectively, different default ports.)
This approach lets avoid the security warnings from browsers in the case when you have insecure content loaded by secure pages (and the pages can be loaded using both http and https protocols).