The centripetal force (definition) is the net force on any object moving in a circular path which has direction towards the center of the circle. Centripetal force is not the new kind of force, it is simply the result of net force given by Newton's second law (\(\sum \vec F = m \vec a\)) which always has direction towards the center of the circular path.
The centripetal force is also called radial force meaning the force is along the radius of the circle. Sometimes you'll see the term radial being used more often. The word centripetal is taken from Greek word meaning "tending towards the center" or "seeking-center". This net force causes the change in direction of velocity of a body moving in circular motion.
When an object undergoes circular motion, two things can happen, one is uniform circular motion and another is nonuniform circular motion. You already know from circular motion that uniform circular motion has only the perpendicular component of acceleration and nonuniform circular motion has both parallel (tangential) and perpendicular components of acceleration.
The centripetal force is caused by the perpendicular component of acceleration called centripetal acceleration or radial acceleration in either type of circular motion, that is we know from circular motion that the centripetal acceleration if an object is moving with speed \(v\) on a circular path of radius \(r\) is
\[a_\text{rad} = \frac{v^2}{r} \tag{1}\label{1}\]
Therefore, you can determine the equation (aka formula) for the the centripetal force on an object of mass \(m\) according to Newton's second law(\(\sum \vec F = m\vec a\)) as,
\[F_\text{rad}=\frac{m{{v}^{2}}}{r} \tag{2} \label{2}\]
If the object completes one rotation in the circle in time period \(T\), it travels a distance equal to the circumference of the circle which is \(2\pi r\). So, the speed of the object is \(v=2\pi r/T\). Now putting the value of \(v=2\pi r/T\) in Equation \eqref{1} we get,
\[{{F}_{\text{rad}}}=\frac{4m{{\pi }^{2}}r}{{{T}^{2}}} \tag{3} \label{3}\]
The above equation gives us the equation (aka formula) of centripetal force or the radial force in terms of the time period of rotation. The examples of centripetal force include swinging object with a string, motion of moon around the Earth, motion of roller coaster etc. In Figure 1 below, you can see a ball is swinging along a circular path with the help of a string. In this case the tension in the string is providing the centripetal force on the ball and hence the ball is moving in the circle.
If that string breaks, there is no centripetal force and the ball can not go along the circle, instead it moves in a straight line at the instant when the string breaks as illustrated by the Figure 2. In case of moon orbiting the Earth, the gravitational force provides the centripetal force on the moon (there is no string in this case).
Since centripetal force is the net force with direction towards the center of the circle, the SI unit is the same as that of net force, that is Newton (N), \(1\text{N} = 1\text{kg}\cdot \text{m}/\text{s}^2\). Note that centripetal force is not the new kind of force.
