Traditionally (in the Black Country and Lancashire) the pudding is boiled and served as a meal in itself, with bread or potato; elsewhere in England (as in Scotland, Ireland, Canada, etc.), it is typically served sliced and grilled as part of a cooked breakfast.

Black Pudding seller at Bury market (source unknown)

Bury Black Puddings
Image: www.buryblackpuddings.co.uk

Stick Black Puddings
Image: www.buryblackpuddings.co.uk
The Southern, or 'stick' form is a sausage of pig's blood with fine oatmeal, minced pork fat and spices forced in c2in diameter skins, made into straight lengths and boiled. There is also a Baked Black Pudding
Early forms of black pudding commonly included milk:
Original Receipt in 'The Accomplisht Cook' by Robert May, 1660 (Robert May 1660);To make blood Puddings
Take the blood of a hog, while it is warm, and steep in it a quart or more of great oatmeal groats, at the end of three days take the groats out and drain them clean; then put to these groats more then a quart of the best cream warmed on the fire; then take some mother of time, spinage, parsley, savory, endive, sweet marjoram, sorrel, strawberry leaves, succory, of each a few chopped very small and mix them with the groats, with a little fennel seed finely beaten, some peper, cloves, mace salt, and some beef-suet, or flakes of the hog cut small. Otherways, you may steep your oatmeal in warm mutton broth, or scalding milk, or boil it in a bag.
Original Receipt from 'A Shilling Cookery for The People' by Alexis Soyer (Soyer 1845)37. Black Puddings, broiled. Make about six or eight incisions through the skin with a knife, in a slanting way, on each side of the pudding; put it on the gridiron for about eight minutes, on rather a brisk fire, turn it four times in that space of time, and serve it broiling hot.
I should recommend those who are fond of black puddings to partake of no other beverage than tea or coffee, as cocoa or chocolate would be a clog to the stomach. In France they partake of white wine for breakfast, which accounts for the great consumption of black pudding. Now really this is a very favorite dish with epicures, but I never should recommend it to a delicate stomach.