There are different types of windmill water pumps. Some use the spinning head of the windmill to raise and lower a piston type of arm that goes up and down to push water towards the surface. Other types of windmill pumps use a screw type pump and as the windmill head turns a series of gears turns a screw type of rotating ring that pulls water towards the surface and into a trough.
On US farms, particularly on the Great Plains, windpumps were used to pump water from farm wells for cattle. In early California and some other states the windmill was part of a self-contained domestic water system including a hand-dug well and a redwood water tower supporting a redwood tank and enclosed by redwood siding (tankhouse).
Early wind pumps directly operated the pump shaft from a crank attached to the rotor of the windmill; the installation of back gearing between wind rotor and pump crank allowed the pump to function at lower wind speeds.
The multi-bladed wind turbine atop a lattice tower made of wood or steel hence became, for many years, a fixture of the landscape throughout rural America. These mills, made by a variety of manufacturers, featured a large number of blades so that they would turn slowly with considerable torque in low winds and be self-regulating in high winds. A tower-top gearbox and crankshaft converted the rotary motion into reciprocating strokes carried downward through a rod to the pump cylinder below. Rising energy costs and improved pumping technology are increasing interest in the use of this once declining technology.
Another method is a chain linked over the windmill drive shaft gear assembly with buckets that are continuously moving along the chain and filling up as they dip into the water and then dump their cargo as they turn over the chain apex.
One of the more common types of windmills use an aerating windmill (one that compresses or pumps air when the head turns) and compressed air from the windmill diaphragm or bellows system goes into a subersible well pump or a floating surface pump and the pressureized pump pushes water out the exit line towards the surface or towards the holding tank.
Some of the more effecient windmill pumps can move up to 1600 gallons per hour or more depending on blade size tower height etc. The more popular form pumps used today can pump from 60 to 200 gallons per hour depending on the head, tubing length and pressure output of the windmill.
The 2" well pump windmill water pump and the horizontal windmill water pump can be seen here, they can also be used with electric air pumps: Windmill Air Pumps
If you want more specific information on windmill water pumps and how they work with various windmill systems a good source, of course is Wikipedia. There are also some great books if you are a history buff and want to see more about windmills:
An interesting History of American Windmill Technology