FAA accredits flight schools by reviewing their equipment, facilities, personnel, and curricula. Individual operators and very small programs usually don't meet the FAA standard for program size, even if they are perfectly legitimate. The FAA makes a directory of certified schools available on its website.
Non-accredited schools and flight training are perfectly legal. The FAA points out that the quality of education of non-accredited school can be perfectly fine. Many are smaller operators with sufficient skill and experience to properly train a pilot. Students at non-accredited programs must undergo 40 hours of flight training before they can test for a license -- five more than required for FAA accredited school students. However, the FAA recommends all students train for 60 to 75 hours, which makes the distinction somewhat moot.
Many important parts of aviation education happen on the ground. Learning the principles of aeronautics and the mechanics of flying begin with a book and often in a classroom. The FAA accredits a number of colleges and universities to train pilots as part of their degree and certification programs. Taking pilot training from an accredited college or university can have the advantage of getting education in theory as well as practical training.
The FAA operates its own college accredited by the North Central Association of Colleges and Schools. The FAA Academy doesn't offer beginning pilot training, but covers more advanced aspects of flight and air traffic. Some classes are open to the public, while many are not. To advance your knowledge and skills as a pilot, the FAA Academy gives you an opportunity to learn directly from the accreditors themselves. Many of the courses can be used toward credits at other colleges and institutions.