A thermal mass is a material that absorbs heat from a heat source, and then releases it slowly. Common materials used as a thermal mass include adobe, mud, stones, or even tanks of water.
Within the context of radiant floors, a thermal mass is generally a thick concrete pad that is heated, either from pipes carrying hot water or from a mesh of electric wires. The concrete heats and cools slowly, which protects the room (and the people walking in it) from quick temperature swings. A further advantage is that you may be able to heat the concrete pad at night when electricity rates are low. The pad will then release its heat during the more expensive daylight hours without needing additional electricity.