Ok, the first point to note is that, in a two-way speaker, the crossover for each speaker is usually electrically independent. Not all crossovers are like this, but most are. Because of this, the manufacturer might put two sets of binding posts on the back of the speaker -- one for the tweeter and its half of the crossover, and one for the woofer and its half of the crossover.
Now, biwiring is where you have one amplifier channel per speaker, and run separate wires to the tweeter and woofer. Opinions vary as to the effectiveness of this.
Bi-amping is where you have two amplifier channels per speaker. Typically, this is one channel from each of two different amps. Both amp channels get the same input signal, but one amp feeds the tweeter (via its crossover section) and one feeds the woofer (via its crossover section).
The above is what is sometimes referred to as "passive" bi-amping -- that is, the signal to the drivers is split using the "passive" crossver in the loudspeaker. Another variant is "active" bi-amping. This is where there is no crossover in the speaker at all, and the amplifiers are connected directly to the drivers, without any intervening crossover components. To separate the frequency ranges, an electronic circuit called an actrive crossover is connected between the preamp and the amps -- one amp gets the tweeter signal only and one gets the woofer signal only.
Technically, this last configuration is the most advantageous of the three. The drawback is that crossovers are complicated things; they do more than just split the signal into frequency ranges, and need to be highly customized for the particular combination of drivers and box that the speaker uses. Off-the-shelf active crossovers do none of this.
Hope this helps
