The precise nature of chemical-bonding interactions in amorphous, and crystalline, chalcogenides is still unclear due to the complexity arising from the delocalization of bonding, and nonbonding, electrons. Although an increasing degree of electron delocalization for elements down a column of the periodic table is widely recognized, its influence on chemical-bonding interactions, and on consequent material properties, of chalcogenides has not previously been comprehensively understood from an atomistic point of view. Here, a chemical-bonding framework is provided for understanding the behavior of chalcogenides (and, in principle, other lone-pair materials) by studying prototypical telluride nonvolatile-memory, "phase-change" materials (PCMs), and related chalcogenide compounds, via density-functional-theory molecular-dynamics (DFT-MD) simulations. Identification of the presence of previously unconsidered multicenter "hyperbonding" (lone-pair-antibonding-orbital) interactions elucidates not only the origin of various material properties, and their contrast in magnitude between amorphous and crystalline phases, but also the very similar chemical-bonding nature between crystalline PCMs and one of the bonding subgroups (with the same bond length) found in amorphous PCMs, in marked contrast to existing viewpoints. The structure-property relationship established from this new bonding-interaction perspective will help in designing improved chalcogenide materials for diverse applications, based on a fundamental chemical-bonding point of view.