A theoretical and computational investigation was carried out in the field of laser-plasma interaction using the Finite Difference Method to study the acceleration of electrons with non-relativistic velocities in a on-magnetized and magnetized collisionless plasmas. First, a (Nd:YAG) laser pulse of 25 fs duration and 5x1015 W/cm2 intensity was assumed in the present study. When this laser pulse was allowed to interact with a stationary electron in vacuum, it was found that the electron is accelerated during the interaction only and returns to stationary state after the laser pulse has passed, in agreement with previous works. Also, the interaction of the same laser pulse with a collisionless plasma at electron density ne= 1x1018 cm-3 was studied It was noticed that the energy of the electron during the interaction has reached a maximum value of ~ 1 keV at laser pulse intensity of 5x1015 W/cm2, while the energy of the electron after the interaction reached ~ 15 eV for the same laser pulse intensity.