Oscillations occur when an object is disturbed from a position of stable equilibrium. The classic case is displacement when there is a force of the form F = -kx where x is the displacement from equilibrium.
In that case the resulting motion has the form x = x0cos(ωt+θ0) where we call x0 the Amplitude of the motion and ω the Angular Frequency of the motion. The angular frequency is related to the period (T) and frequency (f) of the motion by
ω = 2πf = 2π/T.
For a mass m on spring of spring constant k we have ω = √(k/m) and for a simple pendulum of length L we have ω = √(g/L).
Transverse waves involve motions of the matter or field that makes up the wave which are perpendicular to the motion of the wave itself. Examples include waves on the surface of water, waves on a stretched spring, and light waves.
Longitudinal waves involve motions that are parallel to the motion of the wave itself. The most common example of a longitudinal wave is sound in its various forms.
Every point on a wave travels at the same speed, the wave speed v. If the wave is periodic with period T and frequency
f = 1/T
then the distance between equivalent points in the wave, the wavelength λ, is realated to the frequency and velocity by
v = fλ.
The energy in an oscillation or a wave is proportional to the square of the amplitude and it alternates beween being kinetic energy of the motion and potential energy stored in the restoring force.