A mattress topper is a separate, removable layer of padding designed to lie on top of your mattress, underneath your fitted sheet. A mattress topper may be filled with foam, memory foam, gel memory foam, latex, down, wool, or synthetic materials, and its purpose is to add a layer of comfort to an otherwise firm mattress.
People sometimes buy a mattress topper to extend the life of an old mattress. This is a temporary fix—it may add a new layer of pressure relief, but the topper will conform to the topography of the mattress beneath it. If your mattress has body impressions, a topper won't be able to fill them in.
Another common reason people purchase a mattress topper is so that they can buy a firmer mattress than they might otherwise need, and then make it softer with a topper. You can always make a mattress softer, but you can’t make it firmer, and a firmer mattress is likely to maintain its integrity better over time. So to maximize longevity, you might consider buying a firm mattress with minimal padding, expecting that you’ll use it with a topper which you’ll replace several times throughout the life of the mattress.
The term “mattress pad” is often used interchangeably to mean either a mattress protector, which may or may not have a layer of padding, or a mattress topper, whose purpose is to add a layer of comfort. You will definitely want a mattress protector; whether or not you choose a mattress topper as well depends on the mattress you've chosen and your personal comfort preference.