The essence of centripetal force is that, according to Newton an object will move in a straight line with constant speed unless you apply an external resultant force on it. To make an object move in a curve (it doesn't have to be a circle) you need the force acting sideways on it. If that curve happens to be a circle, the sideways force acts exactly in the direction of the center of the circle. (This can be proved mathematically.)
That's all it is. You then need to find something to provide that force. If it's a planet in orbit the force is gravity. If it's a mass on the end of a string it's the tension in the string. If it's a car going round a corner it's the friction between the road and the tyres. If it's a train its the force of the rails on the wheels.