What is the Centripetal force, and how does it keep objects in circular motion?

The Centripetal Force is not a new force which begins to act on an object as it moves in a circle. It is in fact the result of an imbalance of the forces acting on an object i.e the net force. This net force acts at or has a component at 90 degrees to the direction the object is already moving in. 
The force as a result does no work, it does not change the speed but only the direction of the objects velocity. Since it is constantly at 90 degrees, as the objects direction deviates the force then again causes it to deviate it's direction, therefore it's trajectory is described by a circle.
The centripetal force should not be confused with the Centrifugal force, there is no such thing as the centrifugal force. There is no reactive force opposing the centripetal force, to push outwards. Newton's third law does not apply here as the Centripetal force is the netforce, which points towards the centre of the circle, the arc of which is what the object travelling across. 
Whenever, you, in a car or roller coaster, feel getting pushed outwards as you move in a circle is not due to a force pushing you outwards. It is e fact that your body has inertia or mass, and it wants to keep traveling in the forward direction, whilst the car movies inwards, the resistance to the inward force cause you to think there is another force pushing you outward. When in reality you are moving forwards whilst the car is moving inwards 90 degrees to you causing you to then follow the cars trajectory. 
 
The Value of the centripetal force is given by mv²/r , where v is the velocity of the object. And r is the radius of the circle.