The word Bourguignon (pronounced "bore-green-YONE") refers to a recipe that is prepared in the style of the French region of Bourgogne (known as Burgundy in English).
Burgundy is one of France's major wine-making regions, producing both red and white wines — although the wine most people associate with the name Burgundy is red.
As such, a recipe prepared à la Bourguignon will likely feature meat or poultry which is braised in red wine along with mushrooms, onions, and bacon.
A classic recipe prepared in the Bourguignon style is Beef Bourguignon (Boeuf à la Bourguignon in French), which features beef braised in red wine, pearl onions and button mushrooms, flavored with strips of pork fat called lardons.
A Bourguignon sauce is based on the demi-glace sauce.
Traditionally, Beef Bourguignon was made with a large cut of meat from the beef round primal cut, such as the top rump or top round. Since these are leaner cuts of meat (and in those days, beef was much leaner than it is today), the strips of pork fat would be threaded into the meat using a long needle in a process known as larding.
It was basically a way to simulate marbling, and it's pretty much never done anymore. But the recipe still includes cubes or bonnets of salt pork or bacon, which is first browned, rendering the fat which is then used to brown the meat in preparation for braising it.
A quick note on larding: Although it's not necessary for modern beef, which is much more marbled than it was a couple of centuries ago, it is useful for preparing game meats such as venison, which is very lean and dry.
And in any case, when I make Beef Bourguignon, I use beef chuck rather than beef round.
Interestingly enough, the classic French dish is known as coq au vin (rooster in wine) also originated in the Burgundy region and it is essentially Beef Bourguignon prepared with a rooster instead of beef.
Rooster is tougher and gamier than chicken, so it was useful to prepare it by slow cooking with the strong flavor of the wine, although today it's almost always prepared with chicken.
Sometimes you'll see Beef Bourguignon prepared with peas and/or carrots, mainly to add color to a preparation that when prepared along the traditional recipe is rather monochromatic.