Often when you are talking about a standalone ice maker you are really talking about something called "clear ice". Needless to say, these ice makers produce perfectly clear ice that tastes amazing--no white air bubble and no garbage or sediment from your water source.
The way these ice makers work, in general, is to run water over a plate or shoot water into a "bell" that is cooled by a refrigerant. Since the surface is freezing, the running water will eventually begin to freeze in slow layers and the heavy sediment just gets pushed away by the water. So what you end up with is perfectly clear, pure ice.
Rather than just wasting all the water that runs over the plate but doesn't ice up right away, a recirculating pump pushes the water back up over the tray. Once the thermistor by the plate registers that it is cold enough, it reverses the flow of the refrigerant in the system, simultaneously defrosting the system and heating the plate up. When the plate gets hot enough, the ice melts where it is touching the plate and then slides down the incline to a wire grid that melts its way through the clear sheet of ice-creating perfect cubes of amazing clear ice.
In the models that use jets to force the water up into a refrigerated "bell" shape, the same thing happens when the refrigerant is reversed. The bell becomes hot and melts the first layer of ice and gravity drops the ice bells into the bucket--no cutting needed.
So while this process is happening, the recirculating pump sits a little bit higher in a reservoir than the bottom. That way, all of the heavy sediment collects in the bottom and only the clean water gets recirculated. When the "harvest" happens and the refrigerant is reversed, the inlet pump runs the entire time. This causes all of the water in the reservoir to rise and push out all of the old water (along with the sediment and other impurities).
The wastewater is then combined with the water from the melted ice and comes out the back of your icemaker. A gravity-drain icemaker requires a floor drain so that the water can just drain out of the unit without needed a pump.
This is almost always the best plan though some people will argue. The reason it is the best plan is that you are completely protected should you be out of town and the power goes out. The melted ice will simply run down the drain. With a pump-drain system, the water collects in a waste reservoir until it reaches a certain level. At that point, the drain pump is activated and the wastewater can be pumped into a drain (under your sink like the dishwasher drain works well). Needless to say, if the electricity goes out, your ice will overflow the reservoir and flood your room or back up until all of your ice is ruined.
The best units come with a drain pump that has a secondary trigger that turns the interior recirculation pump off if the waste reservoir gets too full (say if your drain pump fails).
I have had the pleasure of owning 3 of these clear ice makers in my life and I have built and rebuilt and combined and recombined them and I cannot emphasize enough how great the ice is every time. Granted it's not nugget ice, but it doesn't cost 6 grand either.