The structural evolution and atomic structure of the Ti50Cu50 compound have been investigated by means of molecular dynamics simulation using the generalized embedded-atom model (GEAM) potential. Gibbs free energy calculation manifests the large driving force of undercooled Ti50Cu50 for crystallization and thus the poor glass-forming ability. Radial distribution functions (RDFs) within the temperature range from 2000 K to 300 K are analyzed and reveal the increasing degree of short-range order and reducing periodic length between peaks on cooling. Atomic arrangement is characterized by the Voronoi tessellation method, showing that the frequency of icosahedral configurations is most sensitive to temperature and grows upon quenching while that of the others remains relatively stable. The thermal behavior of the structure factors follows the Debye model up to the supercooled liquid temperature. The structural investigation of amorphous Ti50Cu50 demonstrates that there exist a variety of polyhedral configurations in Ti50Cu50 amorphous alloy, where icosahedral and bcc clusters are the major types. Due to the existence of bcc clusters and the other distorted polyhedra other than full icosahedra, the structural analysis reconfirms the inference from the Gibbs free energy calculation.