Initially founder James E. Casey's employees delivered the messages on foot, then made some deliveries via motorcycle and later purchased model-T Fords for package delivery. The vehicles were painted different colors so the public would see that they had a fleet rather than just one car.
In 1916 Charlie Soderstrom joined UPS and selected brown for their uniforms and delivery vehicles. He chose a hue of brown that was similar to "the color used on Pullman rail cars because the color reflected class, elegance, and professionalism - and dirt is less visible on brown uniforms and vehicles," according to UPS. By 1929 the color brown that is still used today was adopted across the company.
UPS has recently gone one step further in "owning their color" than most companies by incorporating it into their slogan "What can brown do for you?".
Interesting to me was that while the public had recognized brown as a trademark of UPS it wasn't until 1998 that UPS actually registered official trademarks on the color brown to prevent other delivery service companies from using their signature color.
To read more about the history of UPS visit their corporate website.