The below is all UK-specific:
A private school is one which is not run by the state. It will still be checked by the state, but is far more free to run itself and use its own educational philosophy. Because they receive little to no money from the state, they generally cost money to go to.
A public school is a subset of the private school. It is, in theory, open to anyone. In reality it is open to members of the middle to upper classes who have a great deal of money. The most commonly-accepted current definition of a public school is one whose headmaster attends the Headmasters' Conference, a prestigious group which serves only to define public schools.
A state school is the opposite of the private; free for those in the area. It may or may not have an entrance exam; those which are academically selective are known as grammar schools; this originally meant that they taught Latin grammar.