Considering the simultaneous propagation of multicomponent fields in an isotropic medium, an N-coupled nonlinear Schrödinger system with the self-phase modulation, cross-phase modulation, and energy exchange terms is investigated in this paper. First, via symbolic computation, the Painlevé singularity structure analysis shows that such a system admits the Painlevé property. Then, with the Ablowitz-Kaup-Newell-Segur scheme, the linear eigenvalue problem (Lax pair) associated with this model is constructed in the frame of the block matrices. With the Hirota bilinear method, the bright one- and two-soliton solutions of this system are presented. In addition, the bright multisoliton solutions of the system for N=2 are straightforwardly derived by the linear superposition of soliton solutions of two independent scalar nonlinear Schrödinger equations. Furthermore, through the analysis for the soliton solutions, the corresponding propagation behavior and applications for soliton pulses in nonlinear optical fibers are considered. Finally, three significant conserved quantities, i.e., energy, momentum, and Hamiltonian, are also given.