Any modern car can really be considered a collection of systems that work together to produce the desired effect - namely, forward motion and stable, safe performance. And the heart of many of those systems are the pumps that regulate the flow of fluids and elements through the critical areas of the car. For most cars, it's not an overstatement to say that the car simply can't operate without functioning pumps.
Types of Pumps
As mentioned above, there are several different pumps within any car, and they are responsible for delivering critical elements where they need to go within the engine, to ensure proper functioning. Each type of pump is unique and uniquely designed for the engine it supports.
In pretty much any engine, oil is the fluid that keeps the pistons, crankshaft and rocker arms function while the engine is running. These run very fast, upwards of 3,000 revolutions per minute. As you might expect, that kind of hyperkinetic motion can produce a huge amount of friction, and that's where oil comes in. Oil is the element in an engine that reduces friction between those components and their housing or other components. Without oil, the heat generated by those components will literally warp the metal of the component, and/or cause them to fuse to their housing. Needless to say, this has the effect of turning your engine into a really big paperweight.
The oil pump, then, is responsible for moving oil through the areas where it's needed to keep those components functioning smoothly. There are several different kinds of oil pumps, but they all function the same way - as long as the engine is running, they make sure that oil is sent where it needs to go.
Modern gas-powered cars utilize what's known as an internal combustion engine, whereby gas, air and a spark all come together to create an ongoing series of controlled explosions that move the pistons and ultimately power the cars. If any of those components fails to arrive at the right time, the car simply won't run or even start. In most modern cars, the fuel pump is actually located in the fuel tank, and it's job is to take in fuel from the tank and send it out across the fuel lines to the engine.
Like most pumps, water pumps are pretty simple in nature - water comes in one side, is spun around by a centrifugal mechanism called an impeller, and is shot out the other end. Usually, the water pump is driven by an accessory or serpentine belt, which is itself connected to the crankshaft of the engine. As long as the engine is running, whether its in gear or idling, the water pump is doing its job.
The water pump, essentially, takes in water that's coming in from the radiator, where it's been cooled, and sends it into the engine, where heat is exchanged, cooling the engine and heating the water, which then circulates to the radiator where it's cooled and flows back to the water pump, and the cycle repeats continuously. Without the water pump, the engine will typically overheat, although it may take a while to do so.
Commonly used in automatic transmissions, transmission pumps are also called gear pumps. As opposed to manual transmissions, which typically use a set of levers or cables, automatic transmissions often utilize a hydraulic system to change gears. The transmission pump is usually located in the transmission itself, and it's job is to draw transmission fluid from a reservoir and send it into the rest of the transmission.
Depending on the make and model of the car in question, there may be other pumps used as well, such as windshield washer pumps and power steering pumps, among others. These 'minor' pumps are typically associated with less-critical functions, but their loss can be acutely felt if they fail.
Looking for Trouble
For the most part, pumps are heavily utilized any time the car is running and, over time, they can wear out and either degrade in functionality, or fail altogether. Some common things to look for when inspecting your pumps are to look at the seals around the pumps, and look for any signs of leakage in from the pump or the surrounding area.
Also, there are typically tests for the various pumps, that use specialized gauges, which can tell you about the status of your different pumps. Those tests will usually need to be done by a mechanic.
If you discover a problem with any of your pumps, replace the unit immediately - it won't recover or get better, only worse. But if you do that in a proactive manner, your pumps will last a long time, and save you from much bigger problems in the future.