Homotopy groups are algebraic structures in the field of topology that help classify and understand the shapes of spaces. They are defined using continuous maps from spheres into a given space, capturing information about the space's structure and holes at different dimensions. The first homotopy group, known as the fundamental group, measures loops in the space, while higher homotopy groups deal with higher-dimensional analogs. These groups are denoted as π_n(X), where n indicates the dimension and X is the space being studied. Homotopy groups play a crucial role in algebraic topology, providing insights into the properties of spaces that are invariant under continuous deformations.