We consider a class of nonlinear second-order differential equations with constant delay and investigate qualitative properties of the solutions, namely, global stability of the zero solution, eventually uniform boundedness of solutions, existence of periodic solutions, and existence of a unique stationary oscillation of the considered equations. As far as the technique of the proofs is concerned, we use the Lyapunov–Krasovskii functional method and the second Lyapunov method to prove our main results. We also improve and correct some former results available from the literature. Finally, in some particular cases, we provide three examples as illustrations and applications of the obtained new results. Hence, we make some contributions to the topic of the paper.