What is the oscillation period?

Hint: Recall that oscillatory motion is the kind of motion in which an object vibrates to and fro about its mean position, and such a motion will have a duration after which the motion repeats itself. Determine how you can define this duration keeping in mind that the object always returns back to its equilibrium position. Consider a simple pendulum to simplify your explanation.

Complete answer:
Let us begin by first understanding what oscillatory motion is. Oscillatory motion is usually defined as the to and fro motion of an object about its mean position. The object basically oscillates or vibrates about its mean or equilibrium position. Under ideal conditions the object can be in a perpetual oscillatory motion but this is not the case since ambient resistance and friction dampens the oscillatory motion till the object settles back into its equilibrium position.
Now, the time taken by an object to execute one to and fro motion about its mean position in the absence of any friction is called the oscillatory period. It is the minimum interval through which it completes one cycle of oscillation or vibration.
For example, consider a pendulum as shown in the figure. Suppose that the mean position of the pendulum is A. The pendulum completes one oscillation upon successively travelling through points ABCDA.

The duration in which the pendulum travels once through ABCDA is called the oscillatory period of the pendulum. This is usually given as:
$T = \dfrac{1}{2\pi}\sqrt{\dfrac{l}{g}}$, where l is the length of the string and g is the acceleration due to gravity acting on the pendulum.
Thus, the oscillation period just gives the time taken by an object to execute one oscillation through its mean position, after which it repeats its motion all over again.

Note:
Do not get confused between periodic, oscillatory and harmonic motion. The motion which is repeated at regular intervals is called periodic motion. The periodic motion in which a body moves to and fro about its fixed position is called oscillatory motion. The oscillatory motion in which the restoring force or acceleration is always directed towards the equilibrium position is called harmonic motion. Note that all oscillatory (vibratory) motion are periodic but all periodic motion need not be oscillatory. For example, the revolution of the earth around the sun is periodic but not oscillatory. Also, all periodic motions need not be simple harmonic.