Psyprof1's got an excellent point: we need to keep biwiring and bi-amping separate, as they are two significantly separate concepts. For that matter, I'd suggest we maintain three specific separate concepts.

• Bi-Wiring: The speaker manufacturer has two sets of binding posts on the rear of the speaker, with jumpers that can tie the two together or separate them. The purpose is to allow two sets of speaker cables (or at least four separate conductors) provide separate positive and negative signal paths from the single amplifier channel for both high frequency and low frequency sections of the speaker. The reasons for bi-wiring are to decrease overall resistance (similar to increasing the wire size) and to separate possible intermodulation, generated by one driver, from influencing the other driver. How beneficial the second reason is can be a matter of some debate, but the first reason is pretty clear-cut.
• Passive Bi-Amping: In passive bi-amping, the source signal (from the pre-amplifier) is split and sent to two separate amplifier channels, each of which is then connected to either the high-frequency or the low-frequency section of the speaker. This provides more power for the speaker. It is possible to encounter problems when mixing amplifiers in this scenario because you can get output level differences that can skew the speaker's overal frequency response, so many people prefer to use identical amps for this. Some will argue that simply getting a single larger amplifier is an equally good solution.
• Active Bi-Amping: In active bi-amping, we get into a whole different level of complexity. The output signal from the pre-amplifier is split, just as with passive bi-amping, but after that we send each signal through a different external crossover network that must be custom designed for the speakers being used. After the two (or more) crossover networks, the separate signals are sent to separate amplifiers. The amplifiers' output go to the speaker, where they are connected directly to the speaker driver - there is now no crossover network inside the speaker. There are significant advantages to this approach: locating the crossovers before the amp can reduce the burden places on the amplifiers (especially the high frequency amp), and the crossover network is dealing much lower level signals. The problem is that very few speakers are manufactured and sold this way, so most active bi-amp setups are DIY - and you need to be very knowledgeable of both crossover circuit design and the speakers in question to tackle this sort of DIY project.

Going back to the case of a typical receiver, there are only two channels of amplification - one amp to drive the left and one to drive the right. This makes either form of bi-amping impossible unless you add external amplification, and leaves us with bi-wiring only.