In cycling terms, a cleat is a part of a clipless pedal system designed to keep one's feet attached to the pedals. The cleat is the 'clip' that is screwed into the bottom of the shoe, that is then clipped into a specially designed pedal.
Of the myriad of manufacturers, Look and Shimano are the two most common, and represent two very different designs. Look pedals are plastic, and are supposed to be replaced every few months, however, thanks to their plastic design, they tend to wear 'into' the pedal and sit very solidly in place. Shimano make their pedals out of metal with rubber stand-offs so you can more easily put your foot down, they last a fairly long time (much longer then Look), but thanks to the metal-on-metal design, some people find them to feel fairly 'slippery'.
Cleats wear down with time, changing their float characteristics, this is more a problem with the Shimano design as they are not as readily disposable and replaceable as the Looks.