How to make custom domains like these
http://fb.me http://youtu.be
You buy them from registrars based in Montenegro and Belgium respectively.
Domain name "youtu.be" is already owned by Google. Domain name "fb.me" belongs to facebook.com
You can check if domain name is already registered for example here: http://www.networksolutions.com/whois/index.jsp
Well, just register the domain name you want (something short probably).
If you want a URL shortener (as the http://youtu.be example) you will need to point this domain to a web application that will convert the short form to the original URL. You can download and install a small webapp (e.g. shortenr) that does, need need to write one yourself.