This refers to sex that occurs between two individuals who once had a relationship, whether married or girlfriend-boyfriend, but who are no longer affiliated. Auld lang syne sex occurs when one or both of the people involved enjoyed the sexual aspect of the relationship, even if they broke up or grew apart.
Jeremy and Jessica had their differences, and ultimately divorced. However, they still got together from time to time for some auld lang syne sex. They figured that they had really terrific sex with each other; and, since they had once been married, it was really okay.