Oil coolers are a sometimes-optional component similar in appearance to radiators or intercoolers, and use air from the grille or other source to transfer heat away from the oil.
Cooler oil lasts longer (since heat degrades oil over time) and helps keep the overall temperature of the engine lower.
Oil Flow Route
The oil is pulled from the oil pan, then (usually) through the oil filter, then out of the engine to the heat exchange area of the oil cooler so the heat from the oil is transferred to the air around the oil cooler. Once cooled, it flows through a line back to the engine, often close to where it came out of, and up into the channels within the engine where it would have flowed without a cooler installed.
Temperature Control
While it is good to keep oil from getting too hot, it is often just as important to keep it from getting too cool, too. A thermostatic valve is often part of the oil cooling system, and allows oil to bypass the oil cooler when the oil is below a predetermined temperature. Oil that is cooled too effectively might not be able to boil off water, allowing water to periodically dilute the oil. Oil that is much colder than the intended temperature of the engine (generally about the temperature of the cooling system thermostat) may also cause thermal issues; in the most exteme cases, it may even crack the block or a cylinder head.
Is an Oil Cooler Worth It? by Donut
In this video, Zach Jobe talks about what an oil cooler does and installs one on a modified Mazda Miata.
Heat Exchanger
A heat exchanger allows two different temperature fluids to equalize their temperatures without the two fluids mixing. The most common type of heat exchanger on a vehicle is an oil-to-coolant heat exchanger. Getting the oil close to the temperature of the coolant is an effective oil temperature regulating method because the coolant temperature is already being controlled by a thermostat. This design is becoming favored by original equipment manufacturers because it can be integrated into engine designs instead of needing to be different for every chassis design the engine happens to be fitted into.
The heat exchanger is typically a metal assembly with multiple passageways made out of a conductive material to allow the oil and coolant to flow very close to each other without mixing, but the conductive materials allow the two temperatures to become relatively equal.