Until the time of Einstein, mass and energy were two separate things. In the special theory of relativity Einstein demonstrated that neither mass nor energy were conserved seperately, but that they could be traded one for the other and only the total "mass-energy" was conserved. The relationship between the mass and the energy is contained in what is probably the most famous equation in science,
E = m c 2
Because the speed of light squared is a very large number when expressed in appropriate units, a small amount of mass corresponds to a huge amount of energy. Thus, the conversion of mass to energy could account for the enormous energy output of the stars, but it is necessary to find a physical mechanism by which that can take place.
Einstein himself originally thought that it might be impossible to find a physical process that could realize the potentiality embedded in his equation and convert mass to energy in usable quantities. In the nuclear age, we now know (both for better and for worse) that he was too pessimistic; there are several physical processes that can accomplish this.