Two-dimensional materials have been considered to be promising potential electrodes of metal-ion batteries. Here we explored the application of three graphene-like carbon-nitrogen structures as anode materials in lithium (Li)-ion and magnesium (Mg)-ion Batteries. We first studied the intrinsic characteristic of C2N, C3N, and g-C3N4, explored possible adsorption positions in each structure, studied the diffusion path, energy barrier, voltage profile, and theoretical capacity. Our results show that the theoretical capacities of C2N anode is 671.7 mAhg(-1) for Li-ions and 588.4 mAhg(-1) for Mg-ions, which makes it a promising anode material. C3N is not suitable as anode material. Because of the few absorptions sites, the capacity of g-C3N4 sheet is only 199.5 mAhg(-1) for Li-ions and 319.2 mAhg(-1) for Mg-ions, which makes it unsuitable as anode material but its performance can be much improved when curled into nanotubes. Using Mg-ions instead of Li-ions can reduce the deformation of the material (for C2N) at the maximum concentration or improve the theoretical capacity (for C3N4), lowing the maximum open circuit voltage while improving the diffusion energy barrier.