A numerical investigation of binary droplet collision has been conducted. The complete process of the collision of two liquid droplets is dynamically simulated by solving the incompressible Navier-Stokes equations coupled with the convective equation of the level set function that captures the interface between the liquid and the gas phases. The simulations cover four major regimes of binary collision: bouncing, coalescence, reflexive separation, and stretching separation. For water droplets in air, the numerical results are compared with the experiments by and Ashgriz and Poo [J. Fluid Mech. 221, 183 (1990)] on collision consequences. For hydrocarbon $\left({\mathrm{C}}_{14}{\mathrm{H}}_{30}\right)$ droplets in nitrogen gas, the simulated results are compared in detail with the time-resolved photographic images of the collision processes obtained by Qian and Law [J. Fluid Mech. 331, 59 (1997)] in every collision regime. The present numerical results suggest that the mechanism of a bouncing collision is governed by the macroscopic dynamics. However, the fact that the present macroscopic numerical model is unable to capture the collision regime of coalescence after minor deformation supports the speculation that its mechanism is related to the microscopic dynamics. Furthermore, the transition from bouncing to coalescence collisions has been predicted and agrees well with the analytical model. The mechanism of satellite droplet formation for head-on collision and stretching separation collision is also studied based on the detailed time-resolved dynamic simulation results. It is then confirmed that end pinching is the main cause of satellite formation in head-on collisions whereas the capillary-wave instability becomes dominant in large impact parameter cases. In the case of an intermediate impact parameter, the effects of twisting and stretching due to the angular momentum and the inertia of the colliding droplets are significant for the satellite formation.