Hilbert schemes are moduli spaces of subvarieties of a quasi-projective variety with fixed Hilbert polynomial?. They’re very nice as moduli spaces go, in that they’re actually quasi-projective varieties.
Existence and construction of Hilbert schemes is due Grothendieck (FGA).
The Hilbert scheme of is widely studied in combinatorics and geometric representation theory for its connections to Macdonald polynomials and Cherednik algebras.