The crossover network is the part of the speaker design that separates the different parts of the frequency spectrum and sends it to each driver. Traditionally, crossovers are implemented passively. Capacitors, inductors, and resistors are assembled in a network and installed in the speaker enclosure, after the power amplifier. Because this circuit has no additional power other than the signal, it is deemed passive.
In an electronic crossover design the crossover goes before the power amps, after your pre-amp (or mixer in a PA type system). You need one power amp for each "way" of your crossover-- 2 for 2 way, 3 for 3 way, etc. x2 for stereo. This crossover is typically implemented with resistors, capacitors and integrated circuits, operating at preamp levels. Because this circuit operates with a separate power supply, it is deemed electronic (or active).
The advantages are:
While the high frequency range doesn't use a lot of power, it is subject to a lot of fast transients, i.e., cymbals. It helps a whole lot to have that range be fed separately.
If a this is too much of a restriction, then there are several ways around this:
A somewhat inelegant, but pragmatic solution is to use an electronic crossover approach assisted by additional frequency shaping with passive components (after the power amps). Whereas this does negate some of the advantages of using an electronic crossover, it does maintain most of them and is as easy to do as regular passive crossover design. (Passive crossover design is "easy" in the sense that is done with large parts and clip leads. It is not easy in the design sense when you deal with all the impedance and phase considerations. Just try to get a perfectly aligned 24 db Linkwitz-Riley transfer function with a passive crossover, which you get right out of the box with an electronic crossover.)
Another approach would be to introduce additional electronic frequency shaping devices after each electronic crossover output, like graphic or parametric equalizers. The disadvantage of this approach would be to really damage the phase relationships, having the effect of clouding up the sound.
The elegant approach is to custom design the entire electronic crossover's frequency shaping. The main disadvantage is that this is a lot more difficult to do and requires considerable circuit design experience.
I feel that the advantages outweigh the disadvantages, and that an electronic crossover design is well worth the trouble.
When using an electronic crossover you don't use a passive crossover: the output of each power amp goes directly to each driver. (Well, you should have some protection for tweeters, usually a token crossover that operates a little bit lower than the active crossover does. Example. If you have a tweeter crossed over at 2,000 Hz, then you'd calculate a series capacitor at say, 1,000 Hz connected directly to the tweeter. Normally this cap would have no effect, UNLESS something got hooked up wrong and you were sending low frequencies to the tweeter. I've been directly driving tweeters in my (electronically crossed over) living room setup (100 watts per tweeter) with no protection with no problem. But, I lead a charmed life and never have any unwanted high-power transients going through my amps.
If you are using an electronic crossover for PA use, you might want to consider using a compressor/limiter. Deluxe way to go would be to have limiters AFTER the electronic crossover, before each power amp. That would REALLY cut down on unwanted distortion. Would make balancing levels tricky, though. Probably only should do that if you have a full-time sound person doing the mix that could twiddle and tweak while you are playing. This probably wouldn't help the sound in a living room stereo, though, but MIGHT be valuable on the subwoofer part, if you overload it a lot.
Electronic Crossovers and Compressors/limiters are widely available from any source that caters to musicians. See links on my main speaker page for more information.
If you want a cheap way to experiment, you can also get inexpensive (less than $100) electronic crossovers that are designed to be installed in autosound systems. The disadvantages of these autosound crossovers are 1., they work on 12 volts, meaning that you have to add a 12 volt power supply, and 2., autosound electronic crossovers are usually only 12 db octave. On the plus side, they use RCA connectors, so that you don't have to fool with finding adaptor cables.
For additional information electronic crossover theory and application, Rane has a good collection of technical notes. Scroll down until you see the ones about Linkwitz-Riley crossovers. Note that a lot of these technical notes are written for professional sound contractors and some of them might be hard to understand and aren't really applicable to home stereo systems.