Circular motion involves the constant changing of velocity (even if the speed is constant the direction is changing all of the time). This means that it involves the acceleration of the moving object.
Acceleration is evidence of a force acting. That force is called 
the centripetal force. It pulls the moving object towards the center of the circular orbit. Without this acceleration, the object would move inertially in a straight line, according to Newton's first law of motion.
Circular motion is accelerated motion even if the speed is constant, because the object's velocity vector is constantly changing direction.
The centripetal force increases with the size of the mass and the magnitude of the speed of the rotation. It decreases as the radius of the circlular path travelled increases.
What provides the centripetal force?
- With orbiting bodies it is gravity that pulls the satellite into a circular path.
- With an object on a thread it is the tension in the thread.
- With a cornering vehicle it is the friction of the tyres on the road
The radius of the turning circle
To make an object take a 'tight cirular path' you have to put in a big force - to change the direction a big mass is travelling in takes a big force and to make an object that is travelling fast take a circular path requires a big force.
Take a look at the following video and think of the force required to pull the hammer's mass into a circle...