A bi wired speaker separates out bass and treble crossovers (in the speakers) and you run two speaker cables from a common amp. Many including me think it is rubbish if your cross over is well designed.
The crossover in the speaker uses big capacitiors, resistors and inductors to filter the various frequencies being sent to woofers, mid and tweeter etc and level balance the speaker levels.
A bi amped speaker uses the same crossovers as above but has two amps driving the speakers bass and treble units separately. This means there is less InterModulation distortion (IM). The bass signals use up most of the amps 'headroom' for want of better term quicker than the treble signals. This means that the treble signals which ride along with the bass signals also get distorted normally (see biwiring and ordinary wiring speakers) even though they are not clipping themselves, it is just the action of the bass signals 'effecting' the treble signals. Biamping greatly reduces this IM and ball park is much like significantly increasing you amp power if you had a single bigger amp. There is a 4 dB advantage here.
The crossovers for the above both work at power levels (power rating of speakers), say 10 to 400w. They do however produce significant distortion which at non bass frequencies is all too audible.
Active speakers remove this 'power' crossover. This greatly reduces Total Harmonic Distortion (THD) which the above crossover produces. (x% down to 0.001% THD) A new crossover circuit does the same job to filter the various frequencies being sent to woofers, mid and tweeter etc and level balance the speaker levels. BUT this time, rather than at power level, it does it at line levels (those tiny power signals being sent down the RCA connectors we all love). Then separate amp power the separate drive units AFTER the crossover rather than before in the bi amping. The crossover is BEFORE the amp now rather than after it.
Bi amping reduces IM
Active speakers reduces THD and IM
Bi wiring is tosh
Active speakers are the BEST I know, I would dearly love some new ones but money and young family
.I have been using active speakers with my PC for 15 years now  They have been turned off twice for two moves
Bluesky, Mackie, Genelec produce some stunning speakers for pro (but brilliant for AV / HIFI as well). ATC, NAIM, LINN and other have always advocated this approach but it can get silly money as my local Linn dealer scared me with on Monday. But active is the best technical solution I know, BUT they also sound the best.
Bluesky and many others however make speakers with amps in the £275 to £375 per active speaker, this is generally two channels of amps and speaker. Their performance is awesome but to get the most out you need a sub but most AV enthusiasts have this already but dont realize what they can do. This is the best way forward IMHO and reduces many amplifier boxes as well. Important to many.
Full range signals need significantly more power than we normally use for AV and Hi Fi currently. Most of our full range amps are grossly under powered for full range signals. Cost thing purely. To solve this either use a 450w amp and bigger or hive off the amp grabbing signals elsewhere (bass). An active bass unit (subwoofer) which most of use already in AV accomplishes this and leave the rest free to use the best options. However we still insist in shooting ourselves in the foot in Hi fi and AV by setting our speakers to full range and not using an active sub where ever we can. It is daft and makes no sense other than people dont know what they are doing.
Amps need to do two things:
1. They need to be of the quality we want (distortion and all that)
2. They need to be able to drive the speaker we have, to the level we want in the room we have.
THESE THINGS ARE SEPARATE
Most people cock up either 1 and / or 2. It costs them money. Fortunately like wires, subs and many other issues it is not difficult to do correctly.
