Why use protocol-relative URLs at all?
It's been an oft-discussed question on StackOverflow what this means:
<script src="//cdn.example.com/somewhere/something.js"></script>
This gives the advantage that if you're accessing it over HTTPS, you get HTTPS automatically, instead of that scary "Insecure elements on this page" warning.
But why use protocol-relative URLs at all? Why not simply use HTTPS always in CDN URLs? After all, an HTTP page has no reason to complain if you decide to load some parts of it over HTTPS.
(This is more specifically for CDNs; almost all CDNs have HTTPS capability. Whereas, your own server may not necessarily have HTTPS.)
Answer
As of December 2014, Paul Irish's blog on protocol-relative URLs says:
2014.12.17: Now that SSL is encouraged for everyone and doesn’t have performance concerns, this technique is now an anti-pattern. If the asset you need is available on SSL, then always use the
https://asset.
Unless you have specific performance concerns (such as the slow mobile network mentioned in Zakjan's answer) you should use https:// to protect your users.