A covering space is a topological space that "covers" another space in a specific way. Formally, if X is a topological space and Y is a covering space of X, there exists a continuous surjective map p: Y \to X such that for every point in X, there is a neighborhood where the preimage in Y consists of disjoint open sets. This means that locally, Y looks like multiple copies of X. Covering spaces are important in algebraic topology, particularly in the study of fundamental groups. The fundamental group of a space can be analyzed using covering spaces, as different covering spaces correspond to different ways of "looping" around the space. This relationship helps in understanding the properties of spaces and their paths.