Active pickups require an electrical source of energy to operate and include an electronic preamp, active filters, active EQ and other sound-shaping features. They can sometimes give much higher possible output. They also are less affected in tone by varying lengths of amplifier lead, and amplifier input characteristics. Magnetic pickups used with 'active' circuitry usually feature a lower inductance (and initially lower output) winding that tends to give a flatter frequency response curve.
The disadvantages of active pickups are the power source (usually either a battery or phantom power), cost, and less defined unique tonal signature. They are more popular on bass guitars, because of their solid tone; most high-end bass guitars feature an active pickup. Most piezoelectric and all optical pickups are active and include some sort of preamp.
The main advantages of active pickups are that they are hum and noise free, at least compared to their passive single coil counterparts, as they require no ground wiring to connect to ground, which also may isolate the player from being electrocuted, should the wiring elsewhere, in other equipment such as a guitar amp, have become hazardous in any way.