There are quite a lot of different fluids inside a modern car. I tried to group them by purpose.
Hydraulic Fluids
General use: Hydraulic fluids are used to transfer power from one point to another
Brake Fluid - Based on glycole-ether, mineral oil or silicone oil. Brake fluids are designed to have high boiling points because the brake system is subject to a lot of heat and hydraulic systems need incompressible fluids to work. If the break fluid starts boiling, it turns into gas - which is compressible. In this case the force from the brake pedal is used to compress the gas instead of moving the brake pads, resulting in a brake failure. Brake fluid is transported by the brake cylinder which is driven by your foot and supported by engine vacuum if your engine is running and equipped with a vacuum servo / brake booster.
Power Steering Fluid - In most cases based on mineral oil. Often ATF (see below) is used as hydraulic fluid inside the power steering. Transported by a mechanical pump driven by the crankshaft using a v-belt.
Clutch Fluid - In some cases the clutch fluid and brake fluid are in the same circuit. That means they share the same reservoir and have separate hydraulic cylinders. Even if they don't have the same reservoir, in most cases the clutch fluid itself is just brake fluid. It is transported by mechanic pressure from your leg.
Lubricants
General use: Lubricants are used to minimize friction between moving parts. This leads to (significantly) longer lasting components.
Engine Oil - transported by a mechanical pump inside the engine (driven by crankshaft)
Gear Oil / Automatic Transmission Fluid (ATF) - I dont't know for sure - but I think its transported inside the gearbox by the gears themselves. If the transmission is equipped with an transmission oil cooler it could be transported with some kind of pump.
Differential Oil - transported inside the differential by the differential gears - if the car is equipped with a differential oil cooler it's moved by an electric pump.
Note: Most lubricants serve a double purpose as coolant fluids.
Coolant Fluids
General use: Coolant fluids are used to transport heat away from components where heat is generated towards a radiator where air transports the heat away from the car.
Engine Coolant - Consists of water, anti-freeze additives and anti-corrosive additives, transported by a mechanical pump inside the engine (driven by crankshaft)
AC Coolant - Transported by a mechanical pump driven by the crankshaft using a v-belt.
Air - Transported to the radiators or cooling fins by dynamic pressure if the car is moving of by fans if the car is not. The fans are driven by the crankshaft or by electricity.
Fuels
Diesel - Transported depending on engine type by an injection pump driven by the crankshaft or camshaft. This can be a unit injector or a high pressure plunger pump (common rail direct fuel injection).
Gasoline - Transported by an electric pump mostly located inside the fuel tank.
Autogas / Liquefied Petroleum Gas (LPG) - Transported by an electric pump located inside the tank
CNG Compressed Natural Gas - Needs no pump as it is stored under high pressure.
Washing fluids
- Windshield washer fluid - transported by an electric pump.
Other fluids
Intake Air - Transported inside the engine by suction (naturally aspirated engine) or by a turbine (turbocharched or supercharged engines)
Exhaust Gas - Transported out of the engine by the engine itself
Nitrous oxide (N2O) - Used to increase engine power. N2O is capable of delivering more oxygen into the engine than air allone could. More Oxygen can be used to burn more fuel which results in higher engine power output. N2O needs no pump as it is stored in pressurized bottles.
Water and/or Methanol for water injection or methanol-water injection. These fluids are sprayed into the intake air to cool it down, and therefore reduce knocking and increase engine power. This system is also called anti-detonant injection (ADI) or MW50.
