A Boiled Dinner takes tougher cuts of meat, or stewing chickens, and simmers them in a pot of water with vegetables (so really, it ought to be called a "simmered dinner.")
A Boiled Dinner differs from a stew in two ways. In a stew, all the pieces are cut down into bite-sized pieces that will fit on an eating spoon, and the broth is thickened and served with them. In a Boiled Dinner, the pieces in it all are larger, and the broth is not served.
The meat used can range from corned beef, beef brisket, pork shoulder, a ham bone with some meat on it, or a chicken. Whatever meat is used, it's used whole, in one piece, as you would for pot roast. The low, slow cooking temperature of the simmering water tenderizes the meat . The vegetables should be added after meat is cooked. Before adding the vegetables, skim any fat off the top of the water. The vegetables used might include cabbage, carrots, onions, parsnips, potatoes and turnips, but never any smaller ones such as peas or kernels of corn. If carrots are used, they are used in large chunks, rather than in a dice.
The vegetables are simmered until just tender. In the pot, they will absorb the flavour of each other and the meat. Root vegetables in particular become very mellow and sweet-tasting in a boiled dinner.
When the dinner is ready to serve, the vegetables are left in the pot to stay hot while the meat is removed, placed on a platter, and carved. You then use a slotted spoon to remove the vegetables, leaving the broth behind in the pot, and arrange the vegetables on the platter around the meat.
The dinner is then served from the platter onto plates, and eaten with a knife and fork.
Boiled Dinners were very popular in Ireland, Scotland and on the eastern coast of North America from New England up to Nova Scotia. In New England, salt cod was sometimes used in place of meat. In Vermont, it is flavoured with bay leaves and garlic; neither flavouring is used in Maine. 
In addition to being a low-cost, wholesome meal, Boiled Dinners were very convenient -- they only needed one pot and one burner, not to mention just one plate per person to serve them on.
While English-speaking foodies today look down on the Irish-Anglo tradition of boiled dinners, they will write glowingly about foreign versions. The French make one called Pot-au-feu, the Germans make Tafelspitz, and the Spanish call theirs "cocido espanol." Amongst the poorer Italian classes, boiled beef dinners were very popular. One Italian boiled dinner for the more prosperous classes, called "Bollito misto alla piemontese", has seven types of meat, seven veg and seven dipping sauces.
Avoid overcooking. If adding beets, cook them separately and add them (without their cooking water) at the last minute, so that all the rest of the food in the pot doesn't go pink. Nice additions are a bag of pease pudding cooking off to the side, or dumplings added towards the end of cooking.
Freeze the flavourful broth for future use in a soup or stew.
"The one-plate meal of early New England was the boiled dinner, a twice-a-week joy from the early autumn until the sap rose in the maples. The meal was cooked in the big iron pot swung from the crane and let bubble merrily over the maple log blaze; the pot lid heaved to the rhythm of regular breathing." Clementine Paddleford (1898 - 1967) 
Literature & Lore
In America, a boiled dinner is often known as a "New England Boiled Dinner."
 Alexander, Kelly and Cynthia Harris. "Hometown Appetites." New York: Gotham Books. 2008. Page 126.
 Ibid, page 126.
Boiled DinnersBoiled Dinners; Bollito misto; Cocido Espanol; Oden; Pot-Au-Feu; Puchero; Tafelspitz
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New England Boiled Dinner