The nLab recognizes two distinct usages of the term “projective set”.
A projective object in a category of sets; see also choice object.
A set of real numbers definable by a formula of second-order arithmetic with real parameters (where “real numbers” has some latitude of meaning among descriptive set theorists, referring to structures that are Borel isomorphic to the standard real numbers).
This article will touch upon the latter usage, as that is what is more commonly understood among mathematicians.