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Ball bearings are used where two parts are joined and need to stay fixed together but also allow one part to rotate inside the other part. Wikipedia has an excellent write up and illustration of how they work. The ball bearings are literally small, metal balls, and there is a track (called a "race") that they follow as the pieces rotate and the balls turn.
Ball bearings are used in bikes in the bottom bracket (crank), pedals, hubs (wheels), and headset (stem/handlebars). If these parts are adjusted and greased correctly, the parts can turn very smoothly and freely but be held firmly in place. For example, cranks can turn in a nice circle but not move side to side at all. If the bottom bracket bearings are tightened too much the cranks no longer turn freely, and if they aren't tightened enough the crank arms can wobble side to side a little.
Ball bearings come in all sizes from a tiny pebble size to baseball size. A diameter of 1/8-3/16 inch is roughly the size of most ball bearings in bikes.
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