In general, lots of money. Most do scholarships, so if your kid is really bright you might get one of those (usually reduced rather than no fees), they do this to bump up their examination results. Connections seem less important now than it used to be, but many will give preference to those whose parents or siblings attended.

Officially money, an interview and an entrance exam.

That's not to say that all things being equal you wouldn't stand slightly more chance if you were a member of the aristocracy.

Money. If you can pay, exceptions will be made for everything else.

It helps if you can trace your ancestors back to the Normal invasion (the winning side, obviously). I'm only half joking.