I got my first "house cat" about 17 years ago, so I've had a lot of time to try many different litters, litterboxes and cleaning methods. This is what I've learned so far.
Reasons to keep your litterbox(es) clean:

1. So it won't be stinky. Yes, I'm sure we all know that...keep it cleaned out and it won't be stinky!
2. So your cats won't go outside the box. Cats are clean animals and they want their litterbox to be clean. The cleaner you keep it, the more likely they are to use it.
3. So if they go outside the box, you'll know its not because the box is dirty.
4. You can tell a lot from what's in the box (or not in the box). You can see if your cat has diarrhea, parasites, is passing blood, etc. Or, if your cat isn't going, you definitely need to know that.
5. To Prevent health problems. If your litterbox isn't clean, your cat may "hold it". This can cause constipation and I'm sure wouldn't be good for their kidneys either.
6. To prevent bad habits. Once your cat goes outside the box, he or she will tend to go in the same place. So its better to just prevent it in the first place.
7. To save time. Its a lot easier to take a few minutes a couple times a day to clean out the boxes than to wait until its full and dirty. That is going to take a lot of time to clean up.
8. To save money. There is a lot less waste if you clean out your boxes often instead of waiting and dumping the whole box every day.
9. Because you love your cat. Imagine if you had to wait on someone to clean your toilet for you, and cats noses are so much more sensitive than ours. So just do the nice thing and keep their boxes nice and clean.
I'm sure there are more reasons, but those are the main ones for me. I've used all kinds of litter boxes--self cleaning, hooded, non-hooded....I even paid over $300 once for a Litter Robot. It was okay, until you had to take it apart to clean it and I realized I couldn't see in the box all the time to keep a check on things. I have finally figured out the regular old way of doing it is actually better. Now I use a large plastic storage tub that I cut a hole in for their covered litterboxes and an extra large litter pan that I ordered from Petsmart (or was it Petco?) for the ones without a lid. (I get the tallest I can find too because Strawberry tends to spray over the box if it isn't tall enough!) Mine all seem to prefer the covered one. I'll admit I haven't always been the best at cleaning the box, but I have learned how important it is and if you clean it regularly, it really doesn't take that much time. Once in the morning and once at night doesn't take long. It's just a habit now. I use the method Dr. Pierson talks about here:
http://www.catinfo.org/?link=litterbox. That's where I got instructions to make my covered litterbox too. I have also probably tried every litter you can imagine, but I think a clumping clay litter is best. I actually prefer the generic wal-mart brand to some of the more expensive ones. I've tried World's Best Litter and I liked it, but it is a little more expensive and hard to find where I live. I don't buy scented litter...the cats don't like it and it really doesn't smell good to me either. Sometimes I sprinkle a layer of baking soda on the bottom of the box under the litter, but I'm not sure if that helps or not. The main thing is keeping the box clean.