Classical Pilates is a movement system designed by Joseph Pilates that focuses on strength, flexibility, control, and alignment. Rooted in a precise sequence of exercises, Classical Pilates emphasizes flowing transitions and the integration of breath with movement. Pilates was created by Joseph Pilates but it was not known as Pilates until after Joseph's death in 1967. When Joe created this method he called it "Contrology", the art of control. His method of exercises coupled with spring tension is what makes his method so beautiful. 

If you have a classical teacher they will utilize the whole system. They will bring you to the reformer, cadillac, wunda chair, spine corrector, etc. All of the equipment that a Classical Pilates studio has is there because it is all part of the system. If your studio does not have all the equipment then it would not be considered Classical.