Centrifugal force

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 star 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 coordinate system in which the blades are stationary (called a rotating reference frame). When a transform is made to that coordinate system, a force term appears which points radially outward from the axis of the blades.
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
 <math> \mathbf{F} = \frac{m v^2}{r} \frac{\mathbf{r}}{r} = {m \omega^2} {\mathbf{r}}<math>
(where m is mass, v is velocity, r is radius of the circle, <math> \omega<math> = v / r is the angular velocity, and the r is the vector pointing from the center to the tip)
This force term is a "fictitious" force because it only appears due to a coordinate transformation. The true nonrotating reference frame can always be discerned by an observer as the one in which there is no centrifugal force.
Physics shorthand
The expression 'centrifugal force' is useful to express in a concise way the dynamics of a system. For example a centrifugal governor is a mechanical feedback device for maintaining a particular revolution rate of a machine. The centrifugal governor of a steam engine usually relies on gravity to provide the centripetal force. When the revolution rate of the centrifugal governor increases, a stronger centripetal force would be needed to maintain the same diameter. The gravity provides only so much centripetal force, so the arms of the governor swing out to a wider angle, to a new equilibrium.
Any explanation of dynamics that is given in terms of 'centrifugal force getting stronger (or weaker)' can be reformulated in terms of 'not enough (or too much) centripetal force than would be necessary for dynamic equilibrium'. But usually 'centrifugal force' is good shorthand for explaining what is going on. Another example of this is the expression tidal force. Tidal force is an apparent force rather than an independently existing force, but the expression is useful physics shorthand.
Inertia
When an object is moved in circular motion, inertia manifests itself. 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 force, because force causes change, and inertia opposes change. When an electric car designed to regain energy on decelerating is switched 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.
When examining the effects of rotation from the perspective of an observer rotating along with the system, the action of the centripetal force shows up as an apparent force term acting in a direction radially away from the center of rotation, and this is the manifestation of the centrifugal force. This term is often called a "fictitious force" because it is actually a manifestation of inertia which only appears as a radially outward force when observing the system from within a rotating reference frame, whereas from a nonrotating frame it is simply observed as the centripetal force producing a circular motion. The appearance of the centrifugal force is one argument used in general relativity for the absoluteness of rotating reference frames, in comparison to the relativity of linear reference frames.da:Centrifugalkraft de:Zentripetalkraft ko:원심력 nl:Middelpuntvliedende kracht ja:遠心力 pl:Siła odśrodkowa sv:Centrifugalkraft