In the late 50s, Grothendieck gave a series of Bourbaki lectures called "Technique de descente et théorèmes d'existence en géométrie algébriques" (available on NUMDAM). These were later compiled in "Fondements de la géométrie algébrique" (FGA). He discussed descent, existence theorem in formal geometry, Hilbert schemes, Picard schemes.
Recently, a series of lecture by Illusie, Nisture, Vistoli & others, explaining all this is in modern langage for students was edited into a book "Fundamental Algebraic Geometry: Grothendieck's FGA Explained".
This is a good introduction to many tools used in moduli theory.