Growth of graphene on copper (100) single crystals by chemical vapor deposition has been accomplished. The atomic structure of the graphene overlayer was studied using scanning tunneling microscopy. A detailed analysis of moire superstructures present in the graphene topography reveals that growth occurs in a variety of orientations over the square atomic lattice of the copper surface. Transmission electron microscopy was used to elucidate the crystallinity of the grown graphene. Pristine, defect-free graphene was observed over copper steps, corners, and screw dislocations. Distinct protrusions, known as "flower" structures, were observed on flat terraces, which are attributed to carbon structures that depart from the characteristic honeycomb lattice. Continuous graphene growth also occurs over copper adatoms and atomic vacancies present at the single-crystal surface. The copper atom mobility within vacancy islands covered with suspended graphene sheets reveals a weak graphene substrate interaction. The observed continuity and room-temperature vacancy motion indicates that copper mobility likely plays a significant role in the mechanism of sheet extension on copper substrates. Lastly, these results suggest that the quality of graphene grown on copper substrates is ultimately limited by nucleation at the surface of the metal catalyst.