Gromov-Witten invariants are mathematical objects used in algebraic geometry and symplectic geometry to count the number of curves on a given smooth projective variety. They provide a way to understand how curves can be mapped into a variety, capturing information about the geometry of the space. These invariants arise from the study of moduli spaces of stable maps, which are formalized as maps from curves to varieties. They play a crucial role in string theory and mirror symmetry, linking geometry with physical theories and providing insights into the structure of Calabi-Yau manifolds.