A moduli problem is a functor on schemes, and the questions are whether it is a sheaf for a topology finer than the Zariski topology, whether it is represented by a scheme, and, when it is not, by what kind of geometric object it is represented.
Functors and topologies — the functors of points of \(\AA^n\) and \(\PP^n\), representability by gluing open subfunctors, sites and the étale and fppf topologies, faithfully flat descent, and torsors.
Hilbert schemes, algebraic spaces and stacks — the Hilbert scheme as a representable moduli functor, quotients by étale equivalence relations, and algebraic and Deligne–Mumford stacks.