Definition 
The simplest vibratory system can be described by a single mass connected to a spring (and possibly a dashpot). The mass is allowed to travel only along the spring elongation direction. Such systems are called Single DegreeofFreedom (SDOF) systems and are shown in the following figure,

Equation of Motion for SDOF Systems 
SDOF vibration can be analyzed by Newton's second law of motion, F = m*a. The analysis can be easily visualized with the aid of a free body diagram,
The resulting equation of motion is a second order, nonhomogeneous, ordinary differential equation:
with the initial conditions,
The solution to the general SDOF equation of motion is shown in the damped SDOF discussion. 