Addition of alumina to sodium silicate glasses considerably improves the mechanical properties and chemical durability and changes other properties such as ionic conductivity and melt viscosity. As a result, aluminosilicate glasses find wide industrial and technological applications including the recent Corning® Gorilla® Glass. In this paper, the structures of sodium aluminosilicate glasses with a wide range of Al/Na ratios (from 1.5 to 0.6) have been studied using classical molecular dynamics simulations in a system containing around 3000 atoms, with the aim to understand the structural role of aluminum as a function of chemical composition in these glasses. The short- and medium-range structures such as aluminum coordination, bond angle distribution around cations, Qn distribution (n bridging oxygen per network forming tetrahedron), and ring size distribution have been systematically studied. In addition, the mechanical properties including bulk, shear, and Young's moduli have been calculated and compared with experimental data. It is found that aluminum ions are mainly four-fold coordinated in peralkaline compositions (Al/Na < 1) and form an integral part of the rigid silicon-oxygen glass network. In peraluminous compositions (Al/Na > 1), small amounts of five-fold coordinated aluminum ions are present while the concentration of six-fold coordinated aluminum is negligible. Oxygen triclusters are also found to be present in peraluminous compositions, and their concentration increases with increasing Al/Na ratio. The calculated bulk, shear, and Young's moduli were found to increase with increasing Al/Na ratio, in good agreement with experimental data.