We use Enzo, a hybrid Eulerian adaptive mesh refinement/N-body code including nongravitational heating and cooling, to explore the morphology of the X-ray gas in clusters of galaxies and its evolution in current-generation cosmological simulations. We employ and compare two observationally motivated structure measures: power ratios and centroid shift. Overall, the structure of our simulated clusters compares remarkably well to low-redshift observations, although some differences remain that may point to incomplete gas physics. We find no dependence on cluster structure in the mass-observable scaling relations, T_X-M and Y_X-M, when using the true cluster masses. However, estimates of the total mass based on the assumption of hydrostatic equilibrium, as assumed in observational studies, are systematically low. We show that the hydrostatic mass bias strongly correlates with cluster structure and, more weakly, with cluster mass. When the hydrostatic masses are used, the mass-observable scaling relations and gas mass fractions depend significantly on cluster morphology, and the true relations are not recovered even if the most relaxed clusters are used. We show that cluster structure, via the power ratios, can be used to effectively correct the hydrostatic mass estimates and mass scaling relations, suggesting that we can calibrate for this systematic effect in cosmological studies. Similar to observational studies, we find that cluster structure, particularly centroid shift, evolves with redshift. This evolution is mild but will lead to additional errors at high redshift. Projection along the line of sight leads to significant uncertainty in the structure of individual clusters: less than 50% of clusters which appear relaxed in projection based on our structure measures are truly relaxed.