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For most mobile applications, including tools like log splitters, a simple gear pump is the cheapest way to get positive displacement output. For years, we would install clutch pump kits under the hood of 1 ton trucks, which would add a double v-belt pulley to the front of the engine that supplied a maximum 25 horsepower to a gear pump that would flow around 8 gallons a minute at 1250 psi. The system would include a 12 gallon reservoir and a power-beyond control valve that supplied a mounted crane in the bed, with an auxiliary tool circuit fed through a selector valve. Later, we switched to Stanley power units driven by 16 hp Briggs & Stratton engines that produced largely the same power from a self-contained unit that could be wheeled about a site or mounted permanently in back of the truck, like an air compressor on a utility rig. Your average dump truck has a big one plugged into the transmission through a power takeoff that feeds the main lift cylinder through a control valve operated from the cab. A power tailgate on the back of a pickup is usually controlled by an electric motor-driven gear pump with a momentary switch nearby.
A different kind of pump that operates the same way as a gear pump is a gerotor pump. Instead of two gears inside, one driven and one driving, a gerotor pump has a lobed roller inside a rotor with matching lobes that changes the displacement as it spins, like a Mazda rotary engine or a Harley Sportster oil pump. You can tell them apart at a glance, because the input shaft on a gear pump is offset from center of the housing, while a gerotor pump both parts rotate around the input shaft, which is centered in the housing.
Another version of a gerotor pump is the vane pump, which has opposing vanes that slide in and out of the rotor as it spins, forcing oil from the low pressure input side to the high pressure output.
If you need more pressure and flow than that, as in a personnel lift device or larger crane, you would be looking at a piston pump, often pressure and flow compensated to give the steady flow and pressure required by the application. Some versions, as in large track hoes and bucket loaders or great big mobile cranes, consist of a variable displacement piston pump coupled to a variable displacement piston motor. Of course, the complexity and expense of such systems goes up dramatically.
There are many other types of pumps out there. If you see a big vactor cleaning out storm drains in town, it could be pumping water through a giant cylinder driven by a pump on the auxiliary engine up by the tank that forces oil in and out one side and water the other. The list is long, but most pumps in use today are one of the first few types I mentioned. Hope that helps.
