Do you want to build it yourself or are you looking for plugins to provide that function?
A quick search brought up InviteReferrals that looks like it does what you want (well, the hard part at least, the output on your site is probably something you want to have influence over).
I’m sure there are others, this was just the first that popped up. Please note that I can’t say anything about its quality, and it has few (but quite mixed) ratings.
Building that yourself shouldn’t be too hard either, I’m sure that’d be a nice project to dive deeper into WordPress development. Customizing the Registration Form looks like a good starting point.