There are three main types of litters:

Clumping litter
Clumping litter forms large hard pieces or clumps when it comes in contact with the moisture of pet waste.  The clumps are then scooped from the box, disposed of, and new litter is added to the box.  The rest of the litter is left almost undisturbed, and lasts about a week to a week and a half.  Boxes filled with clumping litter need to be fully emptied every 1-2 weeks, cleaned with Lysol or a similar cleaner (avoid lemon scents, which irritate cats), and new litter needs to be put in.  There are both scented and unscented varieties, which depends on your cat (some will avoid those that have a smell).  Note:  this type of litter should not be used with kittens - if ingested, it will clump in their tummies and cause problems.  Recommended litter of this type:  Arm & Hammer Clumping Litter.

Silica-based litter
Silica-based litter is composed mainly of sand and water.  There are tiny crystals that absorb odor, and liquid waste is absorbed by granules while solid waste must be removed.  Silica-based litters are bio-degradable and will decompose and return to their sandy base.  I've found that this type of litter is the cleanest - it smells nicer than the others, has very little dust, and the wastes are able to be flushed so it eliminates messes.  Recommended litter of this type:  Fresh Step Crystals.

Clay-based litter
Clay-based litter was the first type of litter introduced commercially, and therefore is the cheapest.  Like the others solid waste needs to be removed daily, and the entire litterbox needs to be completely cleaned out and refilled more often because it absorbs liquid waste and odors (see cleaning directions for clumping litter).  This is not the best type of litter for cats with allergies due to the large amount of dust it produces. 

When changing from one litter category to the next, always gradually mix in the new litter with the old to allow your cat to get used to the new smell and texture.  If you find that your cat rejects the new litter and eliminates elsewhere, you have change the litter too quickly.  If you make a more gradual change and your cat still rejects it, it is not the kind of litter for your cat. 

To eliminate extra cat odors, you can choose one of the scented varieties or sprinkle household baking soda in with the litter to absorb odors.  You can also look into purchasing a commercial cat litter deoderizer, available at your local pet store.