Passive crossovers are electronic filters that do not require external power supplies (passive electronic). In their basic essence, coils and capacitors perform the task of filtering the audio signal after amplification and separating the frequencies to the correct driver. They are the most common crossovers for loudspeaker for 2 reasons: they are low cost to manufacture and you only need a single amplifier channel per loudspeaker. The real downside is how components will affect devices upstreams (E.g. amplifier) and lack any flexibility to "adapt" to a changing speaker. This disadvantage causes considerable grief to loudspeaker designers, battling to improve low to medium overall acoustic performance.

Active crossovers, on the other hand, require an external power supply and will separate the signal before amplification. They, therefore, require one amplifier channel per loudspeaker network. (e.g. Low/Mid/High will require 3 amp channels). No more impedance loading interaction from the passive networks, what you design is what you get! Time alignment couldn't be easier and with cost of amplification coming down, active crossovers are winning hands down over passive crossovers. 

Active Terminology 

- Active filters (as opposed to passive) is made up of active electronic components (Op Amp) to build some filters. Beware as this term is sometimes mixed up with "Digital filters"
- Active speaker is a general term to describe a multi-way speaker which has multiple channels of amplification and an active or digital filter to control crossovers.
- Digital filters would typical use a "Digital Signal Processor" (DSP) which will perform filters using a specialized processor, optimized for running the math equations that's required to process the audio signal.