With regards to fabrics there are four principal types here; breathable, neoprene, nylon and rubber.
As for wader styles there are three types; chestwaders, waist waders and thigh/hip waders.
Of these style, you can also have stocking-foot or boot-foot waders. Stocking foot waders have a neoprene boot to which a wading boot, or shoe needs adding. Boot-foot waders have a boot that’s integral to the waders, very much like a wellington boot.