Pot-Au-Feu is a French cold-weather dish, akin to what an English-speaker might call a “boiled dinner.”
The dish has a mythical place in French cooking, particularly on Sundays. There is at least one entire book dedicated to the topic: “Pot au feu” by Julia Csergo, 218 pages long (published by Autrement, 1999.)
Pot-Au-Feu is a sort of stew with lots of relatively clear broth. The meat is usually beef, accompanied by root vegetables. The method in which it is served is very similar to fancier presentations of the Austrian boiled dinner called “Tafelspitz.”
The dish is designed to be at its best after a long, long period of simmering over low heat. Its name literally means “pot in the fire.” Traditionally, it was meant to be made in a pot hanging over the kitchen fire, and just be available ready to eat all day, probably chucking in new ingredients to top it up as you went along.
Pot-Au-Feu is often trumpted as a “a complete meal dish”, but that’s misleading, because some traditional sides such as Parsley Potatoes and Green Sauce can be a bit of extra work.
To make Pot-Au-Feu, you want cuts of beef that are good for stewing such as shin, flank, brisket, topside, silverside, oxtail, etc. Short ribs give particularly good body and flavour. Some say ideally, you use a mix of cuts. Veal can be added. Many recipes call for the addition of chicken parts. Some recipes add sausage, though purists say no form of pork is ever allowed. But there are regional variations: some might well toss in pork and lamb.
Most say a good Pot-Au-Feu should start with stewing bones, which start off the cooking process, going into the simmering water at least an hour before the meat does. Some omit the bones, and say the beef should go into cold water, and you raise the temperature from there — others respond that that is the perfect way to end up with cardboard meat. This reflects an ongoing dispute between what is more important in Pot-Au-Feu: the broth, or the meat.
After the bones have simmered for an hour or so, you then add the beef, let that cook a while, then add sliced vegetables such as onions (whole), leeks, celery, carrots, garlic, turnips, celeriac, etc. (Potatoes are best cooked separately so that they don’t cloud the broth.) Seasoning is usually simple: salt, pepper, cloves, and a bouquet garni. In the south of France, you might see chickpeas and saffron tossed in, as well as some white wine.
You let everything simmer. It should never boil. The slow cooking draws the flavours and gelatines from the bones to give flavour and body to the broth.
Towards the end of cooking, you add bone marrow.
When it is all done, you slice the meat, and arrange it on a platter, long with the vegetables, then cover the platter to keep warm (you can even put it in a slow oven.)
To serve, you first strain the broth, and serve it in bowls with chopped parsley as a garnish (you don’t thicken the broth, as you would for an English beef stew.) Larousse Gastronomique suggests that other items can be added to the broth: “the soup — which should have toasted bread, pasta products, rice and, in general, all garnishings suitable for clear soups, added to it.” 
The next course is the bone marrow, served on toasted bread, with coarse salt on the side.
Then, you serve the platter of meat and vegetables with side condiments. A Pot-Au-Feu is meant to be bland, so typical condiments include more coarse salt, strong mustard, gherkins, mayonnaise, horseradish, etc. Traditional sides include Parsley Potatoes and Green Sauce.
Some believe that Pot-Au-Feu needs continuous skimming to remove surface grease, but that belongs really to the world of more elevated cooking. Fancy versions may even have you boil beef bones the day before for 12 hours and then use that stock the next day to make your Pot-Au-Feu. But then these fancier versions, made in restaurants, will often only serve the broth and nothing else, not deigning to serve the boiled meat.
Pot-Au-Feu à l’Albigeoise
With beef silverside, veal knuckle, salted pork knuckle, dry sausage, goose confit, and cabbage.
Pot-Au-Feu à la béarnaise
The meat consists of a whole chicken filled with a stuffing made from minced garlic, minced parsley, minced shallots, the chopped livers from the chicken, chopped Bayonne ham and sausagemeat (or ground fresh pork.)
Pot-Au-Feu à la Languedocienne
Beef, some chicken, slab of fatty bacon that has been blanched.
The French can debate as much about what to do with Pot-Au-Feu leftovers, as much as they did about how to make it in the first place. The leftovers can be made into a meat pie, particularly “hachis parmentier” (the French equivalent of Shepherd’s Pie), or diced for a cold salad with a vinaigrette. Leftover broth can be frozen to be used as a stock or a soup base.
Even many French will assume that Pot-Au-Feu is a dish so old that it probably pre-dates the Romans. This doesn’t appear to be so. Pot-Au-Feu as a definite, named dish dates from the start of the 1800s. It may have evolved from the dishes called “pot pourri”, and “hochepot.” One of the earliest references to it and descriptions of it is in fact in English, in the 1825 book “French Domestic Cookery combining Economy with Elegance and Adapted to the use of families of Moderate Fortune By an English Physician”; and the next reference may be Carême, in his “L’art de la cuisine française au dix-neuvième siècle”, published between 1833 and 1844. 
 Soups and Broths — Pot-Au-Feu entry. New Larousse Gastronomique. Paris: Librairie Larousse. English edition 1977. Page 980.
 Hertzmann, Peter. Le Pot-Au-Feu. (History and recipe.) 2005. Retrieved August 2010 from http://www.hertzmann.com/articles/2005/pot-au-feu/
Fabricant, Florence. Pot au Feu. New York Times. 5 March 1995.
Peltre, Béatrice. Bony stew sticks to your ribs. Boston Globe. 16 January 2008.
Sampson, Lynne. The many pleasures of pot-au-feu. The Oregonian. 17 March 2009.