Some constraint languages are more powerful than others because they allow us to express a larger collection of problems. In this paper, we give a precise meaning to this concept of expressive power for constraints over finite sets of values. The central result of the paper is that the expressive power of a given set of constraint types is determined by certain algebraic properties of the underlying relations. These algebraic properties can be calculated by solving a particular constraint satisfaction problem, which we call an 'indicator problem'. We discuss the connection between expressive power and computational complexity, and show that indicator problems provide a simple method to test for tractability.