A ring saw is a form of band saw where the band is rigid, rather than flexible. This requires the band to be circular, rather than the band saw's usual oblong of straight runs between two (or three) guide wheels.
Ring saw blades are abrasive rather than toothed. The brittleness of this abrasive coating, and the need to avoid flexure, is why they are made as rings rather than oval bands. As the ring is a circle of constant radius, the blade is not flexed or bent in operation. The blades themselves are not usually rigid and may be bent slightly in fitting, but are not designed to be flexed as they rotate.