centripetal force
The net inward force that keeps a body moving on a circular path: F_c = m·v²/r = m·ω²·r.
Definition
Centripetal force is not a new kind of force but the role any real force plays when it keeps a body moving on a circular path. The inward acceleration needed to maintain a circle of radius r at speed v is v²/r (or equivalently ω²·r), directed toward the centre. Whatever force supplies that inward acceleration — tension in a string, gravity, normal force from a banked curve, electromagnetic attraction — is acting as the centripetal force for that motion.
The formula F_c = m·v²/r is just Newton's second law for uniform circular motion. For a 1 kg ball whirled on a 1 m string at 2 m/s, the string tension must be 4 N directed inward. For a 1000 kg car on a flat 100 m turn at 20 m/s, the friction between tyres and road must supply 4000 N of centripetal force inward — and if the required force exceeds the friction's ceiling (μ·m·g), the car slides off the outside of the turn.
The "centrifugal force" sometimes invoked in everyday language is not a real force in the inertial frame; it is the fictitious pseudoforce that appears when you analyse the motion from a rotating (non-inertial) reference frame. In the lab frame, only the centripetal force acts, and it is entirely accounted for by string, gravity, or whatever the agent happens to be.