  We study how optical galaxy morphology depends on mass and star formation
rate (SFR) in the Illustris Simulation. To do so, we measure automated galaxy
structures in 10808 simulated galaxies at z=0 with stellar masses 10^9.7 <
M_*/M_sun < 10^12.3. We add observational realism to idealized synthetic images
and measure non-parametric statistics in rest-frame optical and near-IR images
from four directions. We find that Illustris creates a morphologically diverse
galaxy population, occupying the observed bulge strength locus and reproducing
median morphology trends versus stellar mass, SFR, and compactness. Morphology
correlates realistically with rotation, following classification schemes put
forth by kinematic surveys. Type fractions as a function of environment agree
roughly with data. These results imply that connections among mass, star
formation, and galaxy structure arise naturally from models matching global
star formation and halo occupation functions when simulated with accurate
methods. This raises a question of how to construct experiments on galaxy
surveys to better distinguish between models. We predict that at fixed halo
mass near 10^12 M_sun, disc-dominated galaxies have higher stellar mass than
bulge-dominated ones, a possible consequence of the Illustris feedback model.
While Illustris galaxies at M_* ~ 10^11 M_sun have a reasonable size
distribution, those at M_* ~ 10^10 M_sun have half-light radii larger than
observed by a factor of two. Furthermore, at M_* ~ 10^10.5-10^11 M_sun, a
relevant fraction of Illustris galaxies have distinct "ring-like" features,
such that the bright pixels have an unusually wide spatial extent.
