Centripetal Force  The Real Force
Whenever an object moves in a circular path we know the object is accelerating because the velocity is constantly changing direction. All accelerations are caused by a net force acting on an object. In the case of an object moving in a circular path, the net force is a special force called the centripetal force (not centrifugal!). Centripetal is Latin for "center seeking". So a centripetal force is a center seeking force which means that the force is always directed toward the center of the circle. Without this force, an object will simply continue moving in straight line motion.
Imagine swinging a rope with a mass attached to the end, around in a circle above your
head:
Compare each of the following animations to the original one in the top left. Look
carefully for what has changed in each one.





The formula for centripetal force is where m represents the mass of the object, v is the speed (magnitude of the velocity) and r is the radius from the center of the circle to the object. A centripetal force ends up being a net force and a net force always causes an acceleration in the direction of the net force. So if the force is center seeking (centripetal) then the acceleration is also centripetal. The formula for centripetal acceleration is . [Notice that if you multiply this by mass (m) you get the formula for centripetal force...that's because a net force is equal to mass times acceleration.]
It is conceptually better to think about the Centripetal force that is calculated from the formula as a requirement. If you meet the requirement, then you have circular motion at the radius and speed used in the formula. If you do not meet the requirement, then the object moves into a larger curve (which requires less force) or defaults into straight line motion (going off on a Tangent).
In this animation, the "sticky" or adhesive forces from the mud to the tire tread are large enough to be the centripetal force required to keep the mud in a circular path as the tire spins.  
In this animation, the tire is spinning faster which means a larger centripetal force is required to keep the mud in the circular path of the tire. The adhesive forces of the mud to the tire are not large enough to meet the requirement. The mud begins to move into a larger circular path but as soon as it is not touching the tread then there is no force (other than gravity) and so the mud continues with the velocity it had at the instant it was no longer touching the tread. (It went off on a tangent). It followed Newton's first law! 
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