The plural noun burnsides denotes thick side whiskers worn with a moustache and clean-shaven chin.

This style of beard is named after Ambrose Everett Burnside (1824-81), a Union Army general during the American Civil War, as The Century Dictionary Supplement (New York, 1909) explained in the definition of burnsides:

A style of beard such as that affected by General Burnside (1824–81), consisting of a mustache, whiskers, and a clean-shaven chin.

The earliest mention of the proper name Burnside applied to facial hair that I have found is from The Evening Telegraph (Philadelphia, Pennsylvania) of Tuesday 15^th May 1866, which gave an account of the trial of “three of the individuals engaged in the robbery of the Cadiz Bank”:

The thieves, before going into Court, were specially careful of their toilet, and from all reports, as a great many ladies were in attendance, were much admired—looked the “gentleman” much more than honest men surrounding—particularly the thief with the “Burnside whiskers,” who was, in point of physical attraction, voted par excellence.

On the pattern of the earlier term side whiskers, and probably because the original reference to the general was forgotten, the two parts of burnsides were reversed, and sideburns was coined to denote the whiskers grown down either side of the face in front of the ears.

The earliest instance of sideburns that I have found is from one of the Personal Paragraphs published in The Paxton Record (Paxton, Illinois) of Thursday 11^th February 1875:

W. M. McMillen’s side-burns, embellished, genial and classical physiognomy, were visible to his friends of this city, last Saturday and Sunday.

Probably because the element burns in sideburns became opaque, the more familiar term sideboards was coined as an alternative; the earliest occurrence that I have found is from A Radical Step in Life, a story published in The Brooklyn Daily Eagle (Brooklyn, New York) of Sunday 12^th November 1882:

“I think I’ll shave off my side whiskers,” said Needletoes, posing before his looking glass, and taking a comprehensive view of his face. “What do you think about it, old fellow?” he asked turning toward his friend, J. Jenkins Fitz Jones, who was lounging in an easy chair behind him.
“Always thought you looked awfully fresh with ’em on,” said Fitz Jones, rolling a cigarette.
“Honest?” asked Needletoes, covering up his sideboards with his hands, and standing a little back to note the effect.
[&c.]

Among the French equivalents of sideburns are:

– rouflaquettes, a feminine word of obscure origin, now only used humorously, first recorded in the sense of kiss-curls in front of the ears in Croquis de dos, from Le Coffret de santal (Paris and Nice, 1873), by the French poet and inventor Charles Cros (1842-88);

– favoris (literally favourites), a masculine word first recorded in Annette et le Criminel, ou suite du Vicaire des Ardennes (Paris, 1824), by the French novelist Honoré de Balzac (1799-1850).

In this sense, favoris used to be followed by en pattes de lapin, in (the shape of a) rabbit’s legs; hence the usual French word for sideburns, the feminine pattes.

The word favoris was also sometimes followed by en côtelettes, in (the shape of) cutlets.

The following photograph and caption are from the Indiana Evening Gazette (Indiana, Pennsylvania) of Monday 18^th December 1978:

Ambrose Burnside is best remembered now as the father of sideburns. But during the Civil War he was a ranking Union general, and perhaps the greatest bumbler in the history of the U.S. military. He later went into politics.