When you passively biamp, you continue to use the crossover network in the speaker. The signal comes out of the pre-amp and you feed two power amps. If power amp number one feeds the tweeter and woofer of speaker number one, and power amp number two feeds the tweeter and woofer of speaker number two, it is called vertical biamping. On the other hand, If power amp number one feeds the tweeters of both speaker number one and speaker number two, and power amp number two feeds the woofers of both speaker number one and speaker number two, it is called horizontal biamping. In both cases, however, it is "passive" because you continue to use the crossover in the speaker.
Because of inefficiencies in speaker crossovers, another type of biamping is "active" biamping. With this, the signal comes out of the preamp into an electronic crossover, which separates frequencies into high and low frequencies. The high frequencies then go to power amp number one and then to the tweeter terminals of both speakers. The low frequencies go from the electronic crossover to power amp number two and then to the woofer terminals of both speakers. Since the separation of the signal into high and low frequencies has already occurred, you do not need the speaker crossover to do the job, so it must be disconnected.
With a precise separation of the signal by the electronic crossover (which does not occur with a speaker crossover), active biamping eliminates a whole bunch of various distortions which can otherwise occur. It also increases your apparent amplifier power, which helps with dynamics, because passive speaker crossovers suck up about 90% of your signal.
In active biamping, not only do you need to buy another amp, but you need to buy the electronic crossover, so it is a bit more expensive to implement. In addition, you need to perform surgery on your speaker by disconnecting the crossover.
It is extremely important to note that going active presumes that the speaker can be modified in this fashion. Many speaker designers don't envision their speaker crossovers being eliminated for active operation, so they are not desgined to be modified in this fashion. It then becomes a crap shoot as to whether it will be an improvement or not. Unless you know for sure that your speakers are meant to be upgraded in this fashion, you probably should not even attempt to go active.