Despite the fact that this particular facial hair style had been around since at least 100 BC (one of the earliest known depictions is a mosaic of Alexander the Great), sideburns were named after a specific man in the late 19th century.
The man was politician, businessman, and Union Army General Ambrose Burnside. Burnside sported particularly prominent facial hair on his cheeks that was connected to a moustache, while keeping his chin shaved perfectly clean. Here's what he looked like:
Burnside was popular both as a General and, later, as a politician. That popularity, in combination with the fairly unique formation of his whiskers, helped start something of a new facial hair trend in the 1870s and '80s. People called the style "burnsides."
Within a few years of this, the style of facial hair down the side of one’s cheeks, sans mustache, began to be called “sideburns"; the first recorded instance was in 1887. (In some regions, they were also called "mutton chops.") The shift came from the fact that this part of the “burnsides” facial hair style was on the sides of the face—and of course, leaving the “burns” part in in homage to the aforementioned style.
Shortly after “sideburns” popped up, an alternate, “sideboards,” also made its debut, with “boards” thought to have been shortened from “border,” so essentially “side-boarder,” which is a fitting description of the style.