This chapter presents the single degree of freedom (SDOF) mass-spring-damper system, a fundamental model in the study of mechanical vibrations. The analysis of its motion plays a crucial role in developing a solid understanding of how engineering structures and machines respond dynamically. The analysis of the dynamics of this system introduces essential concepts that underpin the formulation of more advanced numerical models, which incorporate multiple inertial components—such as masses and rigid bodies—connected through elastic and damping elements. In addition, the chapter includes practical insights and illustrative examples that demonstrate the application of SDOF systems in engineering practice.