There are a number of types of Driving Schools out there, and a number of different ways of categorizing them. There are racing schools, autocross schools, and general high performance schools, with some functional overlap between the different types -- for example, a high performance school won't teach you some racing techniques (starts, passing in corners, etc.) but it will allow you to sample driving at the limit and refine your technique in a controlled environment, and decide whether racing is really for you. There are schools which supply the cars, and schools where you are expected to supply a suitable car. There are professional schools, and there are various Club Schools. In the following discussion, we will talk about each of these issues.
There are two general categories of Racing Schools: those put on by sanctioning bodies such as the SCCA for their members, often at a financial loss, and professional schools such as The Skip Barber Racing School, Russell Racing Schools, and the Bob Bondurant School of High Performance Driving, who are trying to make money offering racing schools. Each type has strengths and weaknesses, and there is also some variation from one to the next. Sanctioning body schools usually expect you to supply personal gear and a race car (either you own it or you rent it), while professional schools generally supply the race car, and most of the necessary personal gear. The price for professional racing schools is correspondingly higher.
High performance schools differ somewhat from Racing schools in that the emphasis is on basic driving technique, as opposed to racing technique. In general, high performance schools work on the "fast line" as opposed to the "racing line" (often not exactly the same thing), prohibit passing in corners, and in general don't permit behavior that looks like racing. Many professional racing schools also offer high performance classes.
There are two classes of high performance schools; one type generally supplies street cars, and uses a relatively small number of instructors who mostly observe from outside the car. The other type generally relies on students to supply cars, and uses a larger number of instructors who ride with the students to provide more direct feedback. The former schools are generally more expensive, and are always for profit, professional operations. The latter schools are cheaper, and are divided between for profit, professional operations and so-called Marque or Club schools. Note that often the Marque schools are very high quality, especially those put on by the Porsche Club of America and the BMW Car Club of America.