We give clear explanations visuals for the path product and homotopies of paths in a topological space. We focus on the fact that the path product is not necessarily associative, but that we can extend the path product to equivalence classes of homotopic paths, and show that the path product is associative on these homotopy classes. Along the way we show that a path is homotopic to any reparameterization of itself. If you like this algebraic topology stuff, let me know in comments!