The noun exhaust refers to what comes out of your car's tailpipe. It consists of the burned gases and particulate matter that are created by the engine. As a verb, the word exhaust means to use something up, such as a food supply or your energy.
The exhaust system in your car is the part that takes the waste created by the car in the form of gas or smoke and expels it through the tailpipe. The noun exhaust appeared in the 19th century, after the invention of the engine. Before that, the word, as a verb, referred only to depleting something, such as a natural resource. Miners deplete a mineral deposit and exhaust the resource.