The main sorts are:

The traditional UK nappy, the terry towel. Cheap, works from birth to potty, fast drying but fiddley to get on and doesn't stay up well by itself. That makes a difference if you don't want your child to always have to wear a vest (bodysuit). Beware that terry towels may get stripped (lint filter) by American style tumble driers, too.

The traditional US nappy is the prefold, much like the terry except slower drying and takes tumble drying better.

Shaped nappies are now very popular, especially from the work-at-home businesses. These are easy to use but don't fit from birth to potty.

With the nappies mentioned above, a waterproof cover needs to go outside the nappy. A pocket nappy has a built in waterproof liner, and it can be stuffed (into the pocket) with most anything, although super-absorbent textiles such as micro-fibre or hemp work best.

There are also wrap-around nappies, All-in-One nappies, fold-out nappies... and new innovations coming onto the market all the time!