There are many cat litter box options available in the pet supply stores - which one you choose will depend on your cat's personality and size. Is your cat timid? Messy? Small or large?
Regular, rectangular open pan (no cover or top):
Covered litter box:
These can either come in two
pieces, with the top and bottom separate, or as a unit. I recommend
the two piece box - if your cat decides the cover is not for them, you
can still use the pan, and clean up is easier.
One word of caution - some covered litter box kits include a swinging door as an option. I tried installing it on one of our boxes to see what our shyer cats would think. Oops. They quickly discovered that if someone on the outside of the box whacks the door with their paw, it will swing back and into the cat using the box. They thought this was the best "hurry up" signal ever. I took it off after a very short time, much to everyone's disappointment. By all means, if your kit comes with one give it a try, but have another litter option available in case your cats think like ours do.
Covered boxes are good for shyer cats who dislike the openness of other cat litter boxes, or who don't want anyone looking at them while they are in there. If you have a smaller apartment and the box is visible, you may want to try a covered box for a shy cat. Some males can also use a covered box if they tend to not "squat" enough and overshoot the edge of the box.
Size is more important in a covered litter box than an uncovered one. Your cat will not want to feel squashed or uncomfortable in the box. If they do, they will probably stop using it. Most cats like to dig and find just the right spot to go. The box must be big enough for the cat to walk in and turn around comfortably. The doorway must also be large enough.
Covered boxes come in multiple sizes and all kinds of shapes. If you have not tried a covered box before, start with the (right sized) plain rectangular style until you know what your cat likes.
There is one that "rolls" from side to side for cleanup - the cat uses it, and the owner rolls the cat litter box to the side to trap the waste in a receptacle. The receptacle is then removable. Good for people who don't like, or can't, scoop out the box. This box needs to be used with clumping litter.
How Many Boxes?
The general rule is one for each
cat, plus one (so 3 cats would need 4 boxes). If you have less, they
will need to be cleaned much more often and there may be territory
problems. It's fine to have a mix of styles to accommodate different