The expression centrifugal force is used to express that if an object is being swung around on a string the object seems to be pulling on the string. In actual fact the person holding the string is doing the pulling. When an object is at speed, then if no force is exerted the object will continue in a straight line. To make the object deviate from that straight line a force must be exerted.

When a stone is being swung around on the end of a rope the tension in the rope is transmitting the force directed to the center that is being exerted by the person swinging the rope. On the other end of the rope the stone is attached and since the stone itself is not attached to anything it cannot resist the force and the direction of motion is bent; towards the center.

With a planet orbiting around a sun the same dynamics are at play. Without any force the planet would move in a straight line. The sun's gravity is bending the motion away from that straight line. Because the planet has a lot of velocity perpendicular to the bending force the distance to the sun doesn't decrease.

In general, the force maintaining the circular motion of an object is called the centripetal force.

Centrifugal force in calculations

When performing calculations, for example on the stresses in the blades of a helicopter, it is convenient to use a model in which the blades are stationary, and to use in the model a force that points in the direction of the tip of the rotorblade.

The force directed away from the center that corresponds to an amount of mass m at a distance r from the center is given by

(where m is mass, v is velocity, r is radius of the circle, ω = v / r is the angular velocity, and the r is the vector pointing from the center to the tip)

Rendering the blades "stationary" is a fictitious model, in actual fact the blades are rotating. In order to make the calculation deliver an answer that is valid for the rotating blades exactly the right fictitious force is introduced.

Inertia

Inertia is a form of resistance to change, in this case change of velocity. Inertia manifests itself in response to acceleration. Inertia does not prevent acceleration, as it depends on it.
Inertia is very different from forces, because forces cause change, and inertia opposes change. When an electric car designed to regain energy on decelerating is swithched to braking, the manifestation of inertia is driving the generators, charging the electric car's battery system. In this example, inertia is exerting a force, but inertia cannot keep the car going: inertia only manifests itself when the velocity is changing.

When an object is moving in a straight line, then to change the direction of motion a force perpendicular to the direction of motion must be exerted. The resulting acceleration in that direction is the same as would have occurred when accelerating from a stationary start: motions that are perpendicular to each other are independent.

In the case of circular motion: as the centripetal force is causing deviation from moving in a straight line inertia is manifesting itself, but it does not prevent the centripetal force from maintaining the circular motion.

In this sense there is no 'centrifugal force'. The centripetal force is being opposed, but circular motion is not prevented. It is better to use the word 'force' only for driving power that can cause acceleration, or that can neutralize another force. When inertia is manifesting itself, it should be called 'manifestation of inertia'