A stagecoach is a type of covered wagon

Covered wagon

The covered wagon, also known as a Prairie schooner, is an icon of the American Old West.Although covered wagons were commonly used for shorter moves within the United States, in the mid-nineteenth century thousands of Americans took them across the Great Plains to Oregon and California...

 for passengers and goods, strongly sprung and drawn by four horses, usually four-in-hand

Four-in-hand (carriage)

A four-in-hand is a carriage drawn by a team of four horses having the reins rigged in such a way that it can be driven by a single driver. The stagecoach and the tally-ho are usually four-in-hand coaches....

. Widely used before the introduction of railway transport

Transport

Transport or transportation is the movement of people, cattle, animals and goods from one location to another. Modes of transport include air, rail, road, water, cable, pipeline, and space. The field can be divided into infrastructure, vehicles, and operations...

, it made regular trips between stages or stations, which were places of rest provided for stagecoach travelers. The business of running stagecoaches or the act of journeying in them was known as staging.
The stagecoach was supported on the thoroughbraces, which were leather straps supporting the body of the carriage and serving as shock absorbing springs (the stagecoach itself was sometimes called a "thoroughbrace").