Structural and electronic properties of neutral and anionic magnesium clusters with 2 to 22 atoms are studied using gradient-corrected density functional theory. A,new scheme for the conversion of the Kohn-Sham eigenenergies into electron binding energies is utilized to compute the difference in the binding energies of the two most external electrons in the anionic clusters. The results are in very good agreement with the data obtained in recent electron photodetachment experiments. The other electronic properties studied include (a) the binding energy, the second difference of the total energies, the HOMO-LUMO gap, and the vertical ionization potential of the neutral clusters; (b) the vertical electron detachment energy of the anionic clusters; and (c) the character of bonding in both the neutral and the anionic clusters. The analysis focuses on the finite-size analogue of the insulator-to-metal transition. The role and manifestation of the finite-size effects are discussed, and some important implications regarding the interpretation of the experimental data are pointed out.