This article studies the problem of finite-time stability (FTS) and finite-time contractive stability (FTCS) for nonlinear impulsive systems, where the possibility of time delay in impulses is fully considered. Some sufficient conditions for FTS/FTCS are constructed in the framework of Lyapunov function methods. A relationship between impulsive frequency and the time delay existing in impulses is established to reveal FTS/FTCS performance. As an application, we apply the theoretical results to finite-time state estimation of neural networks, including time-varying neural networks and switched neural networks. Finally, two illustrated examples are given to show the effectiveness and distinctiveness of the proposed delay-dependent impulsive schemes.