Duck!

Lemme try this. My explanation is probably horrible but this is how I probably don't understand it...

F=ma is one of Newton's laws.

This means that accelerating bodies experience a force (in the direction of acceleration / change of velocity). An object in circular motion is changing direction, therefore changing velocity, therefore it's experiencing acceleration.

An object in circular motion is effectively an object in straight line motion being accelerated orthogonal to it's motion in such a way that it ends up desciring a circular motion.

Because the change in direction is orthogonal to it's motion, therefore change in velocity is orthogonal to it's motion, therefore the force is felt at right angles to it's motion, which is at right angles to the circular motion, which is centripetal force.

Does that sound like a load of made up shit or what?


Duck!

Lobstrosity

As people have said, the "real" force is always the centripital force. That is to say, if you are in an inertial reference frame (i.e. a frame of reference that is not accelerating), there is no centrifugal force; rather there is only a radially-inward force that is producing the circular motion of whatever is rotating. However, inertial reference frames are not always convenient. Sometimes one wishes to move into a frame that rotates along with the moving body such that it appears to be stationary. An example of this is any frame anchored on Earth, such as you, for instance. You don't perceive the Earth as a rotating body because you rotate with it. To you, the Earth is stationary. Your frame of reference is a rotating frame of reference. A rotating frame is not inertial and thus in such a frame one must add in pseudo-forces to keep physics consistent. From a rotating reference frame, there appears to be a radially-outward force and this force is dubbed the centrifugal force. You see, you don't really know you're moving in a circle. From your point of view, you're not moving at all. However, in reality, you are moving in a circle and your inertia doesn't like this; it wants you to keep moving in a straight line. Your own inertia naively appears to you to be a force pulling you radially outward. Also, just for completeness, note that in rotating frames there is also a secondary pseudo-force known as the Coriolis force that has to do with conservation of angular momentum. It seems strange to us because since we don't readily perceive the fact that we're rotating, we don't readily perceive the fact that we actually have angular momentum due to that rotation.