Carbon Fiber frames show up most often in high-end road and race bikes because it has a few distinct advantages over other common materials: a superior strength-to-weight ratio, longer fatigue life, and increased vibration-damping. Carbon bikes tend to cost more, but the price is worth it for the serious cyclist.
Aluminum frames have their own advantages. The first is cost. Aluminum frames are less expensive to make and, as a result, less expensive to buy. Aluminum's other advantage is its stiffness. While this stiffness means you'll feel more road vibration, it also means that every ounce of power is put directly to the road.