We characterize the finite-gain stability properties for hybrid dynamical systems. By defining a suitable concept of the hybrid norm, we introduce hybrid storage functions and provide sufficient Lyapunov conditions for the stability of hybrid systems, which cover the well-known continuous-time and discrete-time stability notions as special cases. We then focus on homogeneous hybrid systems and prove a result stating the equivalence among local asymptotic stability of the origin, global exponential stability, existence of a homogeneous Lyapunov function with suitable properties for the hybrid system with no inputs, and input-to-state stability, and we show how these properties all imply stability. Finally, we characterize systems with direct and reverse average dwell-time properties, and establish parallel results for this class of systems. We also make several connections to the existing results on dissipativity properties of hybrid dynamical systems.