Modern covers are completely different from the old fashioned vinyl rubber pants your grandmother used back then.   All covers in our store are made of polyster or nylon and made waterproof using PUL (polyurethane laminate) or TPU (thermoplastic polyurethanes) on the inside layer.  Aplix (Velcro or hook & loop as some are called) or snaps are used to fasten them onto the baby.   They come in sized (newborn to large), duo-size, or even one size.  They are used with the prefold, contour, fitted, or as a hybrid diaper.