“Demivegetarianism” is a newly-coined term for a little-known, often misunderstood subset of vegetarianism. A demivegetarian is a person who, for the most part, follows a vegetarian diet. However, a demivegetarian is permitted to consume meat products if and only if those foods are among your absolute favorites. This is different for everyone; I personally am completely vegetarian except for fish (I guess you could also call me a pescetarian), while one of my best friends eats gelatin products in addition to zir vegetarian diet.
As some of you may have guessed, the term demivegetarian was based on the ace community’s “demisexual”. I am demisexual in addition to being demivegetarian, and I really like the parallel drawn.