A molecular-dynamics study of the structure and dynamics of monatomic liquids and glasses is presented. The local atomic structure and its development during the quenching process are analyzed in terms of fluctuations of atomic-level stresses and their correlations. This approach extends the basis for the description of the local structure from the usually employed scalar quantity, the local density fluctuation, to a tensorial quantity, the local stress fluctuation. It is shown here that the local stress fluctuations and their spatial and temporal correlations provide a detailed picture of the dynamics of the liquid and of the transition from an ideal fluid to a viscous liquid, and then to a glass. In particular, it is demonstrated that the shear stresses which are spatially uncorrelated at high temperatures become correlated below a temperature, Ts, which is about twice the glass transition temperature. At the same time the dynamic behavior of the liquid, characterized by the diffusivity, viscosity, and phonon states, changes sharply at this temperature. Implications of this apparent structural transition and its origin are then discussed.