centripetal acceleration

Centripetal acceleration is the 'toward-the-centre' acceleration that keeps anything moving in a circle from flying off in a straight line. The word literally means centre-seeking. You feel its effect as the inward tug when a car rounds a tight bend or a merry-go-round swings you around. Without it, there would be no curving at all — objects would simply travel straight, as Newton's first law demands.

Precisely, for something going round a circle of radius r at speed v, the centripetal acceleration has magnitude a_c = v^2 / r, which can also be written a_c = omega^2 r using the angular velocity. It always points from the object straight in toward the centre of the circle, at right angles to the velocity. Its job is to change the direction of the velocity, not its size — in uniform circular motion it swings the velocity around without ever speeding it up.

The formula tells you the two ways to demand more of it: go faster (and because of the square, doubling the speed quadruples the acceleration) or take a tighter turn (smaller r). This is exactly why a sharp bend at high speed is so much more demanding than a gentle curve, and why race tracks bank their corners.