We develop a novel regional stability analysis framework based on the proposed region-parametrized Lyapunov function to study the synchronization of Kuramoto oscillators. A new synchronization definition is introduced and characterized by frequency boundedness and phase cohesiveness, the latter of which requires phase angles of any two connected nodes rather than any two arbitrary nodes to stay cohesive. This definition allows for time-varying natural frequencies and can lead to less conservative synchronization condition. Applying the analysis framework to Kuramoto oscillators, we derive two algebraic synchronization conditions that relate the underlying network topology and system parameters to the synchronization. Finally, we give estimations of the region of attraction explicitly in terms of phase angles as well as the energy function. They require no calculation for critical equilibrium points compared to traditional methods for power systems. The numerical example shows that two synchronization conditions can complement each other for predicting the synchronization.