I see this in baking recipes, but I've never been quite clear on the difference between mixing and "folding."
"Folding" is a more gentle mixing technique than "stirring" and "mixing". Stirring and mixing both denote a more vigorous action.
Folding is usually used for items where something has previously been whipped (such as egg whites or cream) or where tenderness is desired and thus less mixing is advisable (muffins & biscuits).
Folding is usually done with a rubber spatula (for liquid & dry ingredients) or with a wire whisk (often beneficial for whipped cream and egg whites so that the mixture gently incorporates as it falls through the wires).
To "fold" ingredients together: Hold the spatula or whisk in your dominant hand and grasp the far edge of the bowl (side that is away from you) with your non-dominant hand. Turn the bowl towards you with your non-dominant hand while simultaneously scraping around the edge (also toward you) and finish by folding the mixture over on top of itself. Return both hands to the far side of the bowl and follow-up by cutting through the center of the mixture with your utensil and once again folding the mix over on top of itself (again, turning the bowl simultaneously). Alternate scraping around the side and through the middle of the bowl until the mixture is just combined together.
Note that if an airy mixture such as egg whites or whipped cream start to go from soft and billowy to more liquid-like, you are overworking it and need to stop to maintain volume.
In the case of whipped cream and egg whites you typically add a small portion to the heavier mixture and actually DO stir this in. While yes you're losing some of the volume of that portion, it serves to help lighten the heavier batter/mixture. The rest of the whites/cream are then folded in to the mix in several additions.
Sometimes dry ingredients may be sift on top of whipped eggwhites before being folded.
The larger the spatula or whisk, the fewer strokes that will need to be made and the better the results will be.
Mixing is alot more vicious than folding something in. So if I am mixing I tend to go round and round in circles. However if I am folding something in, I tend to take the spoon from the bottom and lift (ie fold) the mixture onto the top again. It is a slower process but does not cause as much air to be brought into the mixture.
Folding is a technique to gently incorporate one ingredient into another. It's primary purpose is not to overmix, because the jostling would disrupt the texture of the dish. It's particularly common with egg-whites; great care is taken to create air-filled, fluffy dishes, and the addition of a foreign liquid, like, melted chocolate, could easily destroy all of that airiness. Folding is often done with a rubber spatula, because of it's soft edges and flat surface. You can do it with a large spoon if you're careful.