We study intrinsic alignments (IA) of galaxy image shapes within the Illustris cosmic structure formation simulations. We investigate how IA correlations depend on observable galaxy properties such as stellar mass, apparent magnitude, redshift, and photometric type, and on the employed shape measurement method. The correlations considered include the matter density-intrinsic ellipticity (mI), galaxy density-intrinsic ellipticity (dI), gravitational shear-intrinsic ellipticity (GI), and intrinsic ellipticityintrinsic ellipticity (II) correlations. We find stronger correlations for more massive and more luminous galaxies, as well as for earlier photometric types, in agreement with observations. Moreover, the correlations significantly depend on the choice of shape estimator, even if calibrated to serve as unbiased shear estimators. In particular, shape estimators that down-weight the outer parts of galaxy images produce much weaker IA signals on intermediate and large scales than methods employing flat radial weights. The expected contribution of intrinsic alignments to the observed ellipticity correlation in tomographic cosmic shear surveys may be below one percent or several percent of the full signal depending on the details of the shape measurement method. A comparison of our results to a tidal alignment model indicates that such a model is able to reproduce the IA correlations well on intermediate and large scales, provided the effect of varying galaxy density is correctly taken into account. We also find that the GI contributions to the observed ellipticity correlations could be inferred directly from measurements of galaxy density-intrinsic ellipticity correlations, except on small scales, where systematic differences between mI and dI correlations are large.