High-Centered describes the bloom form in which the petals at the center of the bloom stand above the outer opened petals, so that the overall shape of the bloom forms an inverted cup. Hybrid Teas are commonly thought to often have high-centered bloom form. Floribundas and some polyanthas and Teas also have high-centered. High-centered bloom form is thought to be derived from Chinese cultivars that were introduced to Europe in the last 18th and early 19th century and used extensively to breeding modern, repeat-blooming roses because those cultivars carried long, pointed buds.
Blooms that first open high-centered usually continue to fully open to another shape. If the bloom has fewer petals, it's likely to open high-centered and then to finish flat once fully blown (open), revealing the center of the bloom. Others open high-centered and finish cupped.