Having the need to feel superior than others by means of intelligence, athleticism, or any other measurable trait.
Caroline always corrects my grammar when we're talking online. Her superiority complex is so annoying and unnecessary because anyone could understand the point I'm trying to convey, but she still goes to the extremes of addressing the smallest of errors. :v
Sean always talks about the plays he made while playing football, even when trying to compliment others for their play. It's just a natural superiority complex that he has, where he feels the need to boast about himself in order to feel accomplished.