A Leslie speaker is a combination of an amplifier with a special configuration of speakers and horns. The most common use of a Leslie speaker is in conjunction with an organ, and most impressively with a Hammond organ. However, Leslie speakers have been used on many other instruments. For example, George Harrison was known to have used a Leslie speaker on his electric guitar for special effect. Other guitar players have too.
The way that a Leslie speaker achieves its characteristic sound is that there is a stationary bass speaker that faces downward into a baffle that is connected to a motor. By causing the rotor to spin, a special sound effect is achieved. Besides the stationary bass speaker, there is a horn that is attached to a rotor so that it can be spun by a motor. In actual practice there is also a dummy counterweight horn attached as well to keep the system balanced. The combination of the spinning baffle and the spinning horn leads to the well known "Leslie Sound".
In general, depending on the exact model of Leslie and how it is configured, the Leslie can be stopped, operated in "Chorale" in which the rotors are spinning slowly, or "Fast" in which they spin at high speed. By changing these speeds you can achieve different effects. The Leslie requires a certain amount of time to "spin up" and to "spin down", and those lags are part of the specialness of the Leslie effect.
While some Leslies have solid-state amplifiers, most musicians prefer the Leslies with the warm tube amplifiers. Different models of Leslies have different power levels of amplifiers in them.
Some of the more common Leslie models include the 122, 122 RV (has reverb), 145, 147, 21H, 22H, and 760 (the Leslie 760 is considered a "portable" Leslie and has a transistor amplifier.) The Hammond Leslie 3300 Rotary Speakeris a Leslie that is readily available.
It is important to match the correct Leslie to the correct organ or other instrument. You can't immediately swap all Leslies from one organ to another without potentially needing to make wiring changes.
Most Leslies are made like a piece of furniture with nice, polished wood. There are, however, some Leslies that are considered "portable" that are covered with Tolex like a guitar amplifier.
Because many people like the Leslie sound, but don't have access to a Leslie, or don't want to carry one around, there are many electronic Leslie simulators. These usually fail to satisfy. As a matter of fact, I have heard many keyboard players say that they would rather have a simulated Hammond organ with a real Leslie than a real Hammond with a simulated Leslie. I would heartily agree with this sentiment.
Because many people like the Leslie sound, new manufacturers have introduced products to fill the void in the marketplace. One such company is Motion Sound and their Motion Sound Pro-3X Amplified Rotary Speaker
I am personally a huge fan of the Leslie speaker. However, Laurens Hammond, who founded the Hammond Organ Company was not a fan. As a matter of fact, he said that the Leslie speaker ruined the sound of a Hammond organ and went to far as forbidding his staff to have Leslie speakers and threatened to not supply dealers with Hammond organs if they also sold Leslies.
Interestingly, the Leslie speaker was probably responsible for selling more people on the sound of a Hammond organ than Hammond's own "Tone Cabinet". The tone cabinet was a traditional tube amplifier with traditional stationary speakers, that sounded, well, traditional and boring. Don't get me wrong, a Hammond organ through a Hammond tone cabinet sounds nice. But – the sound you are really after is a Hammond organ through a genuine Leslie speaker.