Sideburns are named after a general, but the name got turned around on the way.
Ambrose Everett Burnside (1824 to 1881) served as a Union Army general in the American Civil War. Etymonline says the style of long side whiskers and a mustache, with no beard on the chin, was named for him because he wore it. Merriam-Webster adds that his whiskers became a fashion, and men called them "burnsides."
The dates depend on where you look. Wordhistories.net found "Burnside whiskers" in the Philadelphia Evening Telegraph of 15 May 1866, in a report on the trial of a bank robber. That is only the earliest example the author found, not necessarily the earliest use. Etymonline records the singular burnsides from 1875 and the plural from 1878.
The Century Dictionary Supplement of 1909 defined the style as a mustache and whiskers with a clean-shaven chin. That differs from the sideburns most people picture today. Merriam-Webster now defines them as continuations of the hairline in front of the ears.
The reversal came next. Wordhistories.net records "side-burns" in the Paxton Record of Paxton, Illinois, on 11 February 1875. Merriam-Webster also gives 1875 as the first known use, while Etymonline dates sideburns to 1880 and calls it an alteration of burnsides. The sources differ by a few years, and each date is only the earliest example found.
On why the parts swapped, Merriam-Webster says the order was probably flipped by analogy with side-whiskers. Wordhistories.net agrees and adds a further "probably": the general's name had been forgotten, so the parts were reversed.
The same author reasons that once burns meant nothing to speakers, a more familiar form followed. Sideboards is first found in the Brooklyn Daily Eagle of 12 November 1882.
Burnside's reputation is the other half of the story. A 1978 newspaper caption called him "the father of sideburns" and also "perhaps the greatest bumbler" in U.S. military history. That is the caption's opinion, not a historical verdict.