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This work deals with the analysis and protocol design problems of finite-time consensus for multi-agent systems on signed and directed networks. A new distributed protocol with two reciprocal fractional power feedbacks of local consensus error has been proposed to remove the influence of initial states on the consensus time. By introducing a well-structured Lyapunov function, the technical difficulty arising from the asymmetrical Laplacian matrices of directed graphs is circumvented. Furthermore, it is shown that the feasibility of this proposed protocol can be extended to directed graphs with a spanning tree by mathematical induction method. The upper bounds of the finite-time periods are given under the circumstances of directed networks with a spanning tree apart from the strongly-connected ones. This new protocol is then extended to the application of finite-time bipartite formation control of multi-agent systems. Simulations and experiments of bipartite formation of multiple nano-quadcopters are implemented to prove the validity and practicability of this new protocol. In the experiment, a novel fuzzy logic is also introduced to realize the group collision avoidance of all nano-quadcopters. [ABSTRACT FROM AUTHOR]