From a study of 2.7 x 10/sup 6/ J/psi decays with the Mark III detector at the SLAC storage ring SPEAR, we have obtained new measurements of the two-body decays of the J/psi into a vector and a pseudoscalar meson. We present branching ratios for decays to rho..pi.., KK((892), phieta, the previously unobserved modes phieta', ..omega..eta, ..omega..eta', ..omega pi../sup 0/, rho/sup 0/eta, and new upper limits on rho/sup 0/eta' and phi..pi../sup 0/. In the decay J/psi..-->..eta..pi../sup +/..pi../sup -/, we observe rho/sup 0/-..omega.. interference. The collection of measured branching ratios are simultaneously fit to a simple model of J/psi decays which includes strong and electromagnetic amplitudes and allows for violation of SU(3) invariance. Using this model, we calculate the strange- and nonstrange-quark content of the eta and eta' and conclude that (35 +- 18)% of the eta' wave function can be attributed to an additional component, e.g., gluonium or radial excitation. The measurement of the ..omega pi../sup 0/ branching ratio, when combined with that of the ..gamma pi../sup 0/ decay mode of the ..omega.., yields a determination of the ..omega pi../sup 0/ electromagnetic-form-factor ratio Vertical Barf(q/sup 2/ = m/sub J/psi/ /sup 2/)/f(q/sup 2/ = 0) Vertical Bar = 0.038 +- 0.006.