active vs passive speakers

I am looking at the Gradient Revolution speakers and I can't quite grasp what the active version does. In short, what is an active speaker?
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>Another similar speaker design is powered speakers. Here the crossover still operates on speaker level signals and there's a single amp shared by all the drivers.

There's no good reason to make a powered speaker which uses a passive speaker level cross-over unless you don't care about how loud it plays and have a very simple electrical cross-over function implemented with cheap components.

Now that silicon is cheap even portable stereos are being built with active cross-overs (boom boxes which have 3 or 4 conductor cables connecting the speakers).

The reason is that you get the same output level with a lot less heat sink and power transformer (which are the expensive parts of an amplifer) with an active combination and spend less on high-quality cross over components because you can use low power resistors in place of inductors and much smaller capacitors. It's a consequence of power dissipation being V^2/R. If you're going to end up with 30 and 26V peak signals you need a 56V peak (200W into 8 Ohms) power amp with a passive cross-over while 30 and 26V (60 and 40W) or a pair of 30V amps work with a line level cross-over.

The approach is better too. The cross-overs don't change with output level (due to inductance change with voice coil position and resistance which goes up with temperature) and you can do things that just aren't possible with conventional amplifiers and passive cross-overs (the ability to boost output means you can have a reasonable sized speaker which is 90+ dB sensitive at high frequencies but still plays flat to 20Hz where it's less sensitive but that doesn't matter because there's limited musical content in the last octave).
The active Revolution calls for two amplifiers per side: One for the bass section, and one for the mid/high section. The preamp's output goes to the active crossover, and then the active crossover sends the appropriate signal to the various amplifiers.

In the case of the Revolution, one advantage of the active version is that you can independently control the level of the bass system, and can extend it deeper if you like (up to the woofers' excursion limits and/or your bass amplifier's power limits). If you want, you can add multiple woofer modules for greater bass extension and/or loudness.

dealer(including Gradient)/manufacturer
Nothing much to add to the excellent comments above. It is really such a no-brainer from a technical perspective (cheaper and better) that the only drawback is less flexibility to add distortion to create a desired sound. Some active designs include momentary gain reduction if output gets too high and the amp gets close to clipping - so you can't even add distortion if you try your hardest. A very different animal from a passive speaker - in active you get what you get (manufacturer's design) with very little options apart from the usual placement and room treatments. Not a suitable design for those who like to fiddle with their system and change the sound regularly to make their own unique recipe. For example, speaker cables are not just a minor tweak thay actually become totally irrelevant.

In laymans terms I'll add an analogy to Drew's statement: A tweeter is like Tinkerbell flapping her pixie wings - very delicate and requires almost no power. A bass woofer is like dumbo the elephant - huge power is almost always required - it is the reason that audiophiles prefer massive monoblocks. Now hook Dumbo and Tinkerbell together (please no rude Peter Pan jokes) in a passive system and turn up the volume: it is really no suprise that dumbo's flapping and stomping around disturbs Tinkerbell (IMD distortion). Another analogy is to imagine trying to get Tinkerbell to perfectly choreograph with Dumbo (in phase with a close to ideal transient response) - obviously much harder to do if Tinkerbell has to balance on the elephants back.
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so do you need to own four amplifiers...two per side?
so do you need to own four amplifiers...two per side?

For two way main speakers you will indeed need four power amplifiers for an active configuration. You also need a preamp section that will split the line level audio signal between the two power amps for each speaker. In an active speaker this is usually all built into the speaker for covenience - so you just connect your XLR plugs from your source/preamp to the speaker.

What you will not often find inside an active speaker is tubes. Since tubes are microphonic it is generally a bad idea to put them inside a speaker - tube amplification would require the amplifiers installed in an area away from large amounts of vibration (although this could still be an active design in principle)
The other interesting thing to point out here is that it can be much simpler to adjust active speakers for different rooms with their non-uniform environments across the frequency spectrum.

With simple trim pots you can adjust the level of each frequency band independantly without affecting the cross-over function or having to resort to an external equalizer.

Small (< .5dB) changes can be quite significant.