A side piece is a person someone in a long-term relationship has a physical relationship with. Usually the term side piece denotes females, but some women use the term side piece to describe men they have similar physical relationships with.
The term side piece makes clear that the lover is seen on the side, outside of the main relationship. Piece here is a shortening of the slang term for a woman, “piece of ass.”
When it is applied to women, side piece is a more colloquial term for the word mistress.