The force that keeps a bicycle wheel from flying off is called centripetal force.
Centripetal force acts towards the center of the circular path and is necessary for an object moving in a circle to maintain its circular motion. In this case, it is provided by the tension in the spokes of the wheel and the weight of the bicycle and rider.
Centrifugal force is often referred to in the context of feeling as if you are being "pushed" outward when going around a curve, but it is not a real force; instead, it is a perceived effect in a rotating reference frame.