Connecting speakers

Conventional

The 2 terminals of the amp are connected with the corresponding terminals of the speakers.

Bi-wiring

Today a lot of speakers do have 2 sets of terminals bridged by a jumper.
Remove the jumpers and run separate cables from amp to the speaker terminals.

A bit more on bi-wiring.

Bi-amping

Using 2 stereo amplifiers and connect them to the speaker terminals.
You can use one amp to drive the base and another to drive the tweeters
This is called horizontal bi-amping

You can also use one 2 channel amp for each speaker, using 1 channel for the woofer and the other for the tweeter.
This is called vertical bi-amping.

Bi-amping makes amplifier's life a bit easier.
In a 2 way system a single amp faces a high en a low pass filter and 2 drivers.
In case of B-amping one faces a low pass filter and a driver and the other a high pass filter and a driver.
However, each amp delivers the entire audio range and as a consequence, the filter must remain in place.

Bi-wiring and bi-amping are typical for the audiophile community.

The pros took the consequence.

Active crossover

In the eighties the pro-world discovered that you can improve the transparency of speaker systems if you remove the crossover from the power path.
Simply do the crossover on the line out of the pre-amp.

In our opinion, the best method of connecting your speakers is to supply one amplifier per voice coil and eliminate the speaker's built-in crossover. If your speaker is a three-way design, then you must "Tri-Amplify" your speaker. This technique, however, is not for the feint of heart because you must be prepared to modify your speakers and take some measurements. Individual driver efficiency varies. The original speaker crossover equalized these variations. After eliminating the original crossover, these sensitivity variations must be accounted for by your crossover or amplifiers. Another loose end is the fact that some speaker designers deliberately connect the midrange out of phase. For anyone willing to take the trouble to devote one amplifier to each voice coil, the sonic rewards are outstanding. We can recall an unforgettable lesson many years ago when a manufacturer had two versions of the same loudspeaker. One version used the traditional set-up, the other was Bi-Amplified using built-in amplifiers. Even though the amplifiers used for the Bi-Amplified version were nothing special, the self amplified version always sounded better than the passive version, even when driven by our best amplifier using our best speaker wire.

The DJ Society

As you can see in the text above, The DJ Society is talking about bi-amping too.
In general if the audiophile community talks bi-amping they mean driving a passive speaker with two identical amplifiers.

If the pros talk bi-amping they mean connecting the amp straight to a speaker without a passive crossover and doing the crossover between the pre-amp and the power amp.
In audiophile world bi-amping is passive bi-amping an in the pro-world bi-amping is active bi-amping.

More about active crossovers.

Active speaker

Place the active crossover and amps into the speaker cabinet and you have a active speaker.
Often a low power amp is used for the tweeter and a high power one for the woofer.

No deed to use identical ones.

Almost all studio monitors are build this way.

Slowly active speakers targeting the home appears on the market.

More about active speakers.

Powered speakers

Observe that not all speakers with build-in amplification are actives speakers.
You can place the amp inside the box and drive the speakers over a passive crossover.

Reference

An explanation of bi-wiring - Jim Lesurf
Speaker Cables: Bi-wired! - DonH50
Passive Bi-amping - DonH50