This work addresses input-to-state stability (ISS) for hybrid dynamical systems, which combine continuous-time dynamics on a flow set and discrete-time dynamics on a jump set. The main result entails equivalent characterizations of ISS when the right-hand side of the differential equation for the continuous-time dynamics is locally Lipschitz but the set of admissible derivative values, generated by considering all possible input values, is not necessarily convex. Under some mild assumptions, we show that existence of an ISS-Lyapunov function is equivalent to various types of robust ISS for these hybrid systems. As an application, we demonstrate how a hybrid system framework can be used to study some stability properties for continuous-time systems; in particular, we use the derived equivalent characterization results of ISS for hybrid systems to recover and generalize Lyapunov and asymptotic characterizations of input-output-to-state stability for continuous-time systems.