Recently, atomically thin organic-inorganic hybrid perovskites have been synthesized experimentally, which opens up new opportunities for exploring their novel properties in the 2D limit. Based on the comparative density functional theory calculation with and without spin-orbit coupling effects, the stability, electronic structure, and carrier mobility of the two-dimensional organic-inorganic hybrid perovskites MAPbI3 (MA = CH3NH3) have been investigated systemically. Two single-unit-cell-thick 2D MAPbI3 terminated by PbI2 and CH3NH3I are constructed, and their thermodynamic stabilities are also evaluated using the first-principles constrained thermodynamics method. Our results indicate that both 2D MAPbI3 with different terminations can be stable under certain conditions and have a suitable direct bandgap. Moreover, they are also found to have termination-dependent band edge and carrier mobility. The acoustic-phonon-limited carrier mobilities estimated using the deformation theory and effective mass approximation are on the order of thousands of square centimeters per volt per second and also highly anisotropic. These results indicate that 2D MAPbI3 are competitive candidates for low-dimensional photovoltaic applications.