Of course, there are as many variations as there are housewifes and cooks, plus some regional differences. The two basic ones I know:
Old-fashioned warm potato salad
Boil small potatoes, max. egg-sized, in salt water
peel and slice in 1/8" slices while still hot
fine dice/brunoise some bacon and render with a bit of veg oil
add fine diced shallots or onions and sautee until soft
add vinegar (most common white wine vinegar), pepper
dust with a bit of flour
add stock (more common for home cooks from bouillon cubes)
let cook until slightly thickened
check for seasoning (potatoes will absorb a lot)
mix gently with potato slices
let it sit for a moment to let the potatoes absorb the dressing
serve while still luke warm
can also be served cold but slightly warmed is more common and much better
add enough liquid that the potatoes can absorb and the salad is still moist
Classic potato salad (the way my Mom makes it - very simple but the first thing on my wish list whenever visit home)
Boil small potatoes, max. egg-sized, in salt water
peel and slice in 1/8" slices while still hot
heat stock or bouillon, enough for the potatoes to absorb it, and season with salt & pepper
pour over the potatoes and mix gently
make a simple fresh mayonnaise, thin it with a little bit of stock to dressing consistency
add small diced and blanched shallots or onions to the potatoes
add mayonnaise and gently mix
let it sit for a few hours
serve cold
Depending on where you are in Germany, people may also add cucumber, egg or a bologna-like sausage to it, I like the plain version.
Stefan