What are active crossovers? In short, they electronic circuits that divide the audio spectrum up into discrete bands of frequencies and they function in place the passive crossovers found in loudspeakers. The underlying motivation for the switch to an active crossover is the improved accuracy and flexibility of the active crossover holds over the passive crossover.
Loudspeakers represent truly complex impedances, which only an equally complex passive crossovers can match. Thus deriving the required crossover frequency and slope is difficult with a passive crossover and changing a preexisting passive crossover frequency is anything but easy. Active crossovers, on the other hand, remove the large passive components that make up the passive crossover. This is for the good, as the hundred feet of magnet wire that makes up the inductors and the two back-to-back electrolytic capacitors that make up most crossover non-polarized capacitors are not missed: these components are far from ideal. And the power amplifier, once freed from having to work through this dreck, exercises a better control of the loudspeaker drivers. For example, a damping factor of 100 means little, if the passive crossover adds 1 ohm of DC resistance to the mix, thereby decreasing the effective damping ratio to 8.
Active crossovers also allow for a frequency tailoring that would be altogether impossible or at least incur efficiency penalties with a passive crossover. For example, with a network designed by Linkwitz, we can effectively shift the resonant frequency and Q of a loudspeaker driver. Furthermore, a high Q crossover can boost a drooping low frequency response of a low Q speaker while filtering away sub-sonic garbage, such as record warp.