To deep-fry is to cook foods in a large quantity of hot oil or melted fat. There needs to be enough fat so that the food can “swim” around in it. It is like boiling in a way, except instead of using a large quantity of water, you are using a large quantity of fat.
Some refer to regular frying as “shallow frying” to contrast it with “deep-frying.”
Food that is deep-fried should end up soft inside, and crisp and golden-brown outside.
Most health police now get the shudders even thinking about deep-frying — they say the heat turns benign fatty acids into nasty trans fatty acids and advise against it “under any circumstances.” Still, most people love deep-fried food, even if they’ve sworn off it and are spending the rest of their lives just wanting one more good French fry.
Oil/fat is a very efficient carrier of heat from the source of the heat to the food, better than water or air. Many people debate whether oil can boil or not — or whether the bubbles you see are just air escaping from the food being cooked. Both sides of the debate are correct. The air bubbles that you see are indeed air — steam to be precise — escaping from the food. Oil can boil, though, if you get it hot enough. But the boiling point for many oils happens to be past its smoke point, and past its flash fire point. For instance, the smoke point for soybean oil is 495 F (257 C.) Its boiling point, roughly, is 572 F (300 C.) So you’d never actually want to see oil boil in your kitchen.
In England, a pot that goes on the stove top that you use to Deep Fry in referred to as the “chip pan”
To Deep-Fry, the fat being used must get quite hot. The minimum temperature is usually around 350 F (180 C), though that’s a bit low for good chips. The key to good Deep-Frying is often to fry up small batches at once, so as not to lower the temperature of the oil. If the temperature lowers, the cooking time will take longer, and the food will absorb more oil and get greasy.
Electric fryers (aka thermostatic fryers) can be more efficient than stove-top deep frying, because they heat the oil back up faster after one cooking batch is done, so that you can start the next batch sooner. They have an indicator light telling you when the oil has reached the desired temperature, though the thermostats in them are not always accurate. Electric fryers have lids to keep the splattering contained within the pot. Most now have lids that also have a filter built in, to filter the oil smell and keep it in the fryer. It can be time consuming to clean out an electric deep-fryer after use, so some electric ones now have pan inserts that you can lift right out and put in the dishwasher (for the question of whether to rinse the insert first, see the entry on dishwashers.)
When the heating element goes in electric deep-fryers, it’s often cheaper to buy a new fryer than to replace the element.
Many state fairs in America, such as the Minnesota State Fair (held annually in Falcon Heights, Minnesota, next to Saint Paul) are renowned for the cornucopia of Deep-Fried food on offer, usually on a stick.
For re-using and storing oil for deep-frying, see the entry on “frying oil.”
Some researchers have tried to understand why deep-fried food became so popular in Britain as the Industrial Revolution took hold: “Heavy industry and labor demands eating a lot of calories, and fatty food is a good and cheap source. Deep-frying also kills bacteria and viruses, making it a relatively safe food.” 
Literature & Lore
“…every other cooking method transmutes food, delivering it to us in an altered state. Frying strips away only the rawness, and by its quick, deep heat encapsulates the ingredient with all its intrinsic qualities — the juiciness, the taste, the texture — intact. When, after I was married, I began to cook, the first thing I learned to do and do well was frying.” — Marcella Hazan. “Marcella Says: Italian Cooking Wisdom from the Legendary Teacher’s Master Classes With 120 of Her Irresistible New Recipes.” New York: Harper Collins. 2004.
“…oil is a very efficient medium for transferring heat into food. I looked this up in my favourite reference book, Harold McGee’s On Food and Cooking. Convection currents in oil cook meat “intimately”, McGee explains, at a temperature of between 150C and 175C (300F-350F). The heat dries and browns the surface, causing the Maillard reaction, in which amino acids and sugars are rearranged to create new caramelised flavours. That’s why [deep-fried food] tastes so good.” — Brown, Andrew M. Why deep-fried food is hard to resist. London: Daily Telegraph. 26 July 2014.
Specifying the term “Deep Fry” as opposed to “fry” seems to have first been documented by Websters dictionary in 1930. When reading older recipes, you have to figure out if they meant Deep Fry. For instance, Mary Randolph, in the “Virginia House-Wife”, instructed her readers to cook chicken pieces in “boiling lard.”
Bittman, Mark. Hot, Sizzling Temptations, Freshly Fried at Your Stove. New York: The New York Times. 15 December 2004.
 Morrison, David S. and Mark Petticrew. Deep and crisp and eaten: Scotland’s deep-fried Mars bar. London: The Lancet. Vol. 364, No 9452. Page 2180.