The electrospinning process enables the fabrication of randomly distributed nonwoven polymer fiber networks with high surface area and high porosity, making them ideal candidates for multifunctional materials. The mechanics of nonwoven networks has been well established for elastic deformations. However, the mechanical properties of the polymer fibrous networks with large deformation are largely unexplored, while understanding their elastic and plastic mechanical properties at different fiber volume fractions, fiber aspect ratio, and constituent material properties is essential in the design of various polymer fibrous networks. In this paper, a representative volume element (RVE) based finite element model with long fibers is developed to emulate the randomly distributed nonwoven fibrous network microstructure, enabling us to systematically investigate the mechanics and large deformation behavior of random nonwoven networks. The results show that the network volume fraction, the fiber aspect ratio, and the fiber curliness have significant influences on the effective stiffness, effective yield strength, and the postyield behavior of the resulting fiber mats under both tension and shear loads. This study reveals the relation between the macroscopic mechanical behavior and the local randomly distributed network microstructure deformation mechanism of the nonwoven fiber network. The model presented here can also be applied to capture the mechanical behavior of other complex nonwoven network systems, like carbon nanotube networks, biological tissues, and artificial engineering networks.