Potatoes (preferably a starchy kind, since you want them to thicken the broth), leeks, olive oil go into a large pot and cooked until they soften. In goes stock or water (or a combination) to cover. You bring that to a boil and let it simmer until the potatoes are breaking up and getting really soft. At this point, you're pretty much done. I pureed the soup at this point, but it's not necessary, I just prefer it that way.
Oh, and I added sausage. That move, while perfectly alright, didn't really add much to the soup, and I'd probably leave it out next time.
Served it with some homemade croutons (i.e. a piece of toast I cut up), salad with ricotta salata on the side. Delicious.