Geometric and electronic properties of the endohedral fullerene H2＠C60 and its dimmer have been studied using generalized gradient approximation of density functional theory. It is found that the two hydrogen atoms exist in the form of a molecule in the center of the cage for both H2＠C60 and its dimmer, and the hydrogen molecule can rotate freely at room temperature. The electronic structures of both H2＠C60 and its dimmer show that the formation of hydrogen molecule only affects the energy levels below the Fermi level from -8 to -5eV, while the other energy levels and the energy gap between HOMO and LUMO hardly have any change.