The incompressible magnetohydrodynamics system with variable density is coupled by the incompressible Navier-Stokes equations with variable density and the Maxwell equations. In this paper, we study a new first-order Euler semi-discrete scheme for solving this system. The proposed numerical scheme is unconditionally stable for any time step size $\tau>0$. Furthermore, a rigorous error analysis is presented and the first-order temporal convergence rate $\mathcal O(\tau)$ is derived by using the method of mathematical induction and the discrete maximal $L^p$-regularity of the Stokes problem. Finally, numerical results are given to support the theoretical analysis.