What it is
Mono- and diglycerides are emulsifiers, the quiet helpers that keep oil and water from separating so bread stays soft and ice cream stays smooth. Each molecule is a backbone of glycerol carrying one or two fatty acids, and both of those parts come from fat. Manufacturers make them by reacting glycerol with fats and oils, or with fatty acids, which the US regulations say may be any that are 'derived from edible sources'. That phrase is the whole problem for a vegan reader, because edible fats include soya, palm and sunflower oil but also beef tallow and lard. When European regulators re-evaluated E471 they listed exactly those raw materials side by side, from coconut and rapeseed through to tallow and lard, with nothing on the packet to separate them.
Why the label will not tell you
You will find E471 across the bread aisle, in margarine, in cheap ice cream, in biscuits and snack bars, and in the little tubs of coffee whitener. On a European label it hides behind the code E471 or the phrase 'mono- and diglycerides of fatty acids'; in the United States it is usually just 'mono- and diglycerides'. Neither version tells you whether the fat was pressed from a plant or rendered from an animal, and the two are chemically almost identical once processed, so you cannot tell by reading harder. In practice most large food makers now use palm or soya oil because it is cheaper and simpler to source, but most is not all, and nothing on the packet promises it. The only ways to be sure are to pick a product carrying a registered vegan trademark, or to email the manufacturer and ask which fat they use for their E471.