We present structural measurements for the galaxies in the 0.05<z<0.0585 groups of the Zurich Environmental Study, aimed at establishing how galaxy properties depend on four environmental parameters: group halo mass M_GROUP, group-centric distance R/R_200, ranking into central or satellite, and large-scale structure density delta_LSS. Global galaxy structure is quantified both parametrically and non-parametrically. We correct all these measurements for observational biases due to PSF blurring and surface brightness effects as a function of galaxy size, magnitude, steepness of light profile and ellipticity. Structural parameters are derived also for bulges, disks and bars. We use the galaxy bulge-to-total ratios (B/T), together with the calibrated non-parametric structural estimators, to implement a quantitative morphological classification that maximizes purity in the resulting morphological samples. We investigate how the concentration C of satellite galaxies depends on galaxy mass for each Hubble type, and on M_GROUP, R/R_200 and delta_LSS. At galaxy masses M>10^10 M_sun, the concentration of disk satellites increases with increasing stellar mass, separately within each morphological bin of B/T. The known increase in concentration with stellar mass for disk satellites is thus due, at least in part, to an increase in galaxy central stellar density at constant B/T. The correlation between concentration and galaxy stellar mass becomes progressively steeper for later morphological types. The concentration of disk satellites shows a barely significant dependence on delta_LSS or R/R_200. The strongest environmental effect is found with group mass for M>10^10 M_sun disk-dominated satellites, which are ~10% more concentrated in high mass groups than in lower mass groups.