do some searches and you should find some good info on this.
passive crossovers are those little boxes that come with component sets. they split which signals the mids and tweets play through combinations of small capacitors, inductors, and resistors. the passive crossover receives the full musical signal, and as the electricity passes through the components, they passively split off what goes to what.
active crossover is a box with its own power. you run the signal in, but get to choose at what point the speakers are crossed over at. some simple active crossovers just have knobs. more sophisticated ones have a digital screen and a button to push so you know exactly where you're setting. some are built into what we refer to as a processor, which is a combination eq, crossover, time alignment and usually phase adjustment.
benefits of passive is you plug it in and go. the manufacturer has figured out a good average, and the crossover usually also helps even out any bad areas on the response graph.
benefits of active are flexibility. people that go for these have usually been around long enough to know the risks that come with them...if you make a speaker play frequencies lower than it's designed for, it'll blow. another down side is each speaker needs its own amplifier channel. so where you could use a 2 channel amp with passive crossovers to power a pair of mids and a pair of tweets, you'd need a 4 channel amp to power the same speakers.
