As I understand it, all of these terms refer to cooking food in a small amount of fat/oil. What exactly is the difference?
Pan frying means letting the food sit in the pan and occasionally stirring or flipping. It tends to be done with larger pieces of food, and at a medium to medium-high heat.
Sautéing means shaking the pan back and forth - making the food "jump", if you're translating directly. It's done at a high heat, for a short time, usually with thinly-sliced or finely-chopped ingredients.
Shallow frying, according to some references, refers to the food being partially (halfway) submerged in hot oil and flipped once, as opposed to deep-frying where the food is fully-submerged the whole time.
I prefer the answer in wikipedia:
Sautéing is a method of cooking food that uses a small amount of fat in a shallow pan over relatively high heat
And it differs from pan-frying:
Sautéing is often confused with pan-frying, in which larger pieces of food (for example, chops or steaks) are cooked quickly, and flipped onto both sides. Some cooks make a distinction between the two based on the depth of the oil used, while others use the terms interchangeably.1 Sautéing differs from searing in that searing only cooks the surface of the food. Sautéing is also different from stir-fry in that all the ingredients in the pan are cooked at once, instead of serially in a small pool of oil.
I've never heard of shallow frying...
Pan-frying : Cooking in a very minimal amount of oil with partial coverage.
Shallow frying : Food is only partly submerged about halfway up to the side of the food to be cooked and it must be flipped in between.It is usually used to prepare cuts of fish meat, and for fritters.
Deep frying : Here food is completely submerged in hot fat or oil.
Stir frying : The food is stirred and tossed out very rapidly using wooden or metal cooking utensils.
I wrote a blog post with more about this : Food frying methods
All three frying techniques are different. It is different wrt the amount of oil/fat used.
In deep-frying, the oil should cover the whole food. In short, the food should float in the oil. Similar to french fries.
In shallow frying, the oil should cover your food till half. But, your food should touch the base of the vessel in which you are frying.
In pan-frying, you have to use very little oil. Just enough so that the oil only touches the base of your food.
Sauteing is similar to pan-frying. The only difference is that the oil should be on medium to high heat. The food has to be cooked on small pieces until they all turn brown from the outside and are cooked throughout.