Nappies/Nappy (UK) are traditionally made of terry towelling, a fabric with a 'Nap'.
Nap refers to the raised pile made during the weaving process - threads that stand up from the surface.
They are squares of terry towelling, folded in half diagonally into a triangle, the longest side placed around a baby's waist with a crossover, the remaining point of the triangle passes up between the baby's legs to the waist crossover and secured there with a nappy pin, a large 'safety pin' with a sliding lock to prevent it from becoming unfastened accidentally.
Diapers (US) most commonly refer to shaped padded garments, shaped similarly to the UK nappy but having a more generous cut for the legs, secured at the sides with adhesive strips though other fastenings may be used. They are made mainly of cellullose I understand, some with a filling of gel to absorb urine.
We do have diapers in the UK, colloquially known as Disposable Nappies and in fact they are the most common protection now, but still referred to as nappies. Presently the trash disposal authorities are promoting a swing back to terry towelling because of landfill shortage but citing cost benefits. Professional launderers offer cleaning services. Nursing authorities promote the use of terry towelling nappies too, citing that they are kinder to the baby's skin than cellulose.