The sheer volume of cat litterbox choices can overwhelm any consumer, but especially a first-time cat owner. These are the basic types.
Open: This low-tech, time-tested favorite is filled with litter, either with or without a liner, depending on your preference and your cat's. Some liners are scented; some are designed to stand up to claws. Many brands of open cat boxes have their own brand of liners. These boxes are economical, relatively easy to clean (especially with a liner or clumping litter) and easy for your cat to use.
Hooded: Hooded boxes put a roof over an open box. Bashful kitties may appreciate the privacy; fastidious owners may like the clean look. Odors are also somewhat contained. Cleaning is simple; just pop off the lid. Tentlike versions of box covers are also somewhat contained. Cleaning is simple; just pop the lid off first. There are tentlike versions of box covers, as well.
Motorized: Simply put, they do most of the work for you. You replace waste containers when they get full. Motorized boxes cost more, but the convenience is worth it to many cat owners. And cats that appreciate a clean box like them, too.
Disposable: New to the market, these may not seem economical since they're used once and thrown away, but they're convenient, easy to dispose of and come in handy for traveling or in situations where sharing a box is not an option.