Foam mattresses are made mostly from latex or polyurethane foam. They have no springs, because foam itself provides weight support.
Most of them are made from different foam layers of different density. Foam varies in quality, and mostly determines the quality of all mattress.
In this article we will discuss polyurethane foam mattresses, latex mattresses are in another article.

Pros of foam mattresses:
Relatively easy to clean/transport because of light weight and smaller dimensions.
Less expensive alternative to good innerspring mattresses.
Good recovery properties.

Cons:
Polyurethane is flammable. Rarely manufacturers use flame retardants additions to the foam.

How to choose:
Check its firmness, because they differ.
While buying don't forget to take good warranty, because foam with bad quality can crack / crumble and sag in several months.
Take a look on latex as support material.

Manufacturers and world known brands:

Most distinctive brand with well known marketing history is Tempur-Pedic.

Conclusion:
Comparing to other types, foam mattress is a good choice if it suits your needs and comfort feelings.