This kind of pseudo-force is not limited to circular motion (although it is only called centrifugal force for the case of circular motion). Any object in an accelerated reference frame experiences it. Suppose for instance that you were in a rocket ship so far from any stars or planets that there was no gravitational force on you. Now suppose that the engine of the rocket is burning so the rocket ship is accelerating forward. You are in the rocket ship so you must be accelerating forward too. You will feel a pseudo force pushing you toward the back of the rocket. Indeed, from your point of view, unless you were strapped in when the rocket started accelerating you would feel as if you were falling toward the back of the rocket ship. The pseudo-force would actually be your own natural tendency to maintain a constant velocity. This force would feel very real to you. It would feel just like a gravitational force directed rearward in the ship. To you, the rearward direction would feel like the downward direction does on earth. In fact, if the ship happened to be accelerating forward at 9.80 m/s2 it would feel to you just as if the rocket ship were standing on the earth.
If the rocket ship had no windows you couldn't tell the difference. You would feel as if you were standing on the floor of a rocket ship standing on the earth. Since you would not be moving relative to the things in the rocket ship you would think you were at rest. The pseudo-force pulling you to the floor would feel just like the earth's gravitational force. The floor pressing upward on the soles of your shoes would seem perfectly natural to you. To you it would be the normal force exerted upward on you, equal and opposite to the earth's gravitational pull on you (your weight) and hence of just the right direction and magnitude to keep you at rest in the rocket ship, standing on the floor. Actually, that normal force, exerted on you by the back wall (which you think of as the floor) of the rocket ship, would be the force accelerating you forward through space with the rocket ship. There is no weight. The normal force would be totally unbalanced. That's why you would be accelerating through space with the rocket ship. What you think of as the weight would be the effect of your inertia. You would think the equation
N = mg'
applies where g' is a gravitional field strength, m your mass, and N the normal force; when actually the equation
N = ma
applies where a is your acceleration in the forward direction, what you would think of as upward. The normal force is the same in either case so your pseudo-force, which would feel like your weight, would be equal to your actual mass times acceleration.
mg'=ma
The effective gravitational force constant g', the one you feel, would be equal in magnitude to the actual accleration of the rocket (and you).
g'=a
Of course, with the rocket accelerating forward, the effective gravitational field would be backward, making you feel that the backward direction was downward.
Exotic as this may seem, the sensation is nothing new to you. Anytime you accelerate forward in a car, you feel a pseudo-force pushing you toward the back of the car. The sensation is a result of your inertia and can be thought of as an effective gravitational force in the backward direction. In this case, the earth is still pulling you downward. Conceptually, one combines the real downward gravitational field with the effective backward gravitational field to form a total effective gravitational field in the downward-and-backward direction.
Note that in all cases, the effective gravitational field is in the direction opposite to the direction of the actual acceleration. This holds true in the case of uniform circular motion. The object undergoing uniform circular motion is experiencing an actual centripetal acceleration directed toward the center of the circle. The pseudo-force known as the centripetal force is in the exact opposite direction, namely, outward, away from the center of the circle.