Hilbert schemes are mathematical constructs used in algebraic geometry to parameterize families of algebraic varieties. Specifically, they allow mathematicians to study collections of points in a given variety, capturing information about their configurations and relationships. These schemes generalize the concept of a Hilbert polynomial, which encodes the dimension and degree of a variety. By organizing points into a structured space, Hilbert schemes facilitate the exploration of geometric properties and intersections, making them a powerful tool in the study of moduli spaces and intersection theory.