A self-referral in Google Analytics is a session where the source is your own site. For example, on Distilled.net in September 2015, we had a bunch of sessions show up in Google Analytics like this:
This is often ignored or considered innocuous, but it represents something very wrong with the sessions it represents. It doesn’t make sense for a user to have arrived on your site via your site - there must have been an original source, and we want to include every subsequent hit after that original landing as part of one big session, providing the user doesn’t leave and come back via some other channel, or go inactive for a long period of time (neither of which would appear as self-referrals).
Back in June, I wrote a post discussing the various ways in which Google Analytics can split sessions, with highly misleading results, and self-referrals are often a symptom of this.