Active, or as we usually call them - 'Powered' subwoofers have a amp built in. Generally the driver (speaker) is matched with the amp to provide the best sonic output of low frequency (bass) in a stereo or home theater system.
Also since most modern AVR's (audio/video receiver) only have line-level LFE outputs (RCA style cable to subwoofer) they don't provide an amp to power the sub (like they do for the other speaker channels) so it makes it very easy to add a powered sub to a system. You can still find some 'passive' models that you'd also need an amp to drive it and aren't as common as powered models anymore.
Passive subs are mostly used in car settings where you can mix and match driver size (and enclosure box), ohm ratings, wattage, etc. to a specific amp that best meets the needs of the listener & car.
Hope this info helps.