What exactly is a “three-part drama serial”?

A three part drama serial can refer to any drama in three parts which is meant to be watched sequentially but most often it refers to a drama series on TV (where programming restrictions mean that longer works have to be split into smaller parts). In the theatre (and in the cinema and on TV), plays are often split into acts with intervals between. Such a play, performed on one night, would be called a 'three-act play', not a 'serial'. "3-part" means, of course, that it consists of three programs, usually broadcast on separate days (or, perhaps, on the same day in consecutive weeks).