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:
And it differs from pan-frying:
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