Only two? Well, fundamentally. There are a few companies making beautiful, firm all wool mattresses, similar to a futon, and of course you have the Nest Build a Nest mattress kits, where you can make your own natural mattress like a store bought mattress but using natural and unconventional materials.

But yes, fundamentally, there are two types of foams available, latex and memory foam.

Comparing the two, however, can be a bit like comparing apples and oranges. While both are foams and both are utilized in mattresses, that is as far as the similarities go. 

The first fundamental difference between latex foam and memory foam is the material used to create the foams, and how that dictates their performance. Latex foam is simply rubber. Tree sap, or milk, is dripped into a bucket. It's formed into a block of foam with air injected into it to create tiny pockets or voids, creating a block of flexible rubber foam. And rubber bounces, or deflects, energy. Everyone is familiar with a rubber ball or tire. When you throw it against the ground, it bounces. Same goes for latex rubber used in mattresses. Your body has weight, gravity turns your weight into energy, pushing down into the latex mattress, and the latex rubber in the mattress pushes a good percentage of that energy back at you. 

Memory foam, or synthetic foam, on the other hand, is made from synthetic materials, or plastics, and does not bounce like rubber. In fact, it absorbs energy. Here is a cute little video I made to demonstrate. Don't laugh! 

The item bouncing is a heavy roll of printer paper. The first surface is a pure latex 6" core. The second surface is a memory foam mattress. See how the first surface not only bounces higher but also bounces all over the place while the foam absorbs the falling object and it stays near the area of impact. Latex pushes your energy up at you and also tends to transfer partner movement, while the foam does a better job of absorbing both your energy and partner movement.

There are other differences between the two materials, of course, but that illustrates the key difference in the feel and performance of latex and memory foam.