Periodic Motion is motion that repeats itself in a predictable amount of time (earth revolves around the sun in 365 days or clock arm swings back and forth). This time is known as the period (T). Frequency (f) is the total number of oscillations in a unit of time and the unit is 1/s or Hertz (Hz). Simple harmonic motion is a type of periodic motion where the restorative force is proportional to the displacement.
Systems that are undergoing Oscillations (back and forth motion or repetitive motion) would be at equilibrium if it weren't for being disturbed or displaced. Imagine a pendulum, it would hang motionless, except that its been pushed to one side or another. As it is, a restorative force pushes it back the other way. As it swings past the equilibrium point and the other way, the restorative force disappears and an opposite restorative force begins to work. A spring with a weight can do the same thing compressing and stretching beyond its equilibrium point. The maximum displacement of a swing or stretch (or compression) is called the amplitude.
As a system oscillates, if it slowly loses some of its motion on each period it is said to be damped. If it slowly loses its mechanical energy as it oscillates back to equilibrium it is called underdamped. If it loses too much energy before it actually oscillates it has past a critical damping point and is overdamped. The oscillations of a damped system can be kept going by driving them (driven oscillations), ie adding energy back into the system.
When two oscillations are meshed into a balance that is called resonance (more on this with sound).