Achieving Einstein's Dream
Albert Einstein spent several years of his life trying to develop a theory which would relate electromagnetism and gravity to a common "unified field." Hence the name unified field theory.
Unified Field Theory:
Any theory which attempts to express gravitational theory and electromagnetic theory within a single unified framework; usually, an attempt to generalize Einstein's general theory of relativity from a theory of gravity alone to a theory of gravitation and classical electromagnetism. (McGraw-Hill Dictionary of Scientific and Technical Terms, Sixth Edition, © 1989 McGraw-Hill, New York)
After Einstein finished his first article on the unified field theory in 1922, despite criticism he spent much of the second half of his life pursuing the development of the unified field theory besides the discussion of completeness of quantum mechanics. In the first several years, he was very optimistic, thought success would come soon, but he found it was full of difficulties afterwards. He considered mathematical tools in being was not sufficient, then turned to study mathematics, but never obtained any result with real physical sense. Because Einstein wanted to found an encompassing mathematical construct that would unite not only gravitational field but also electromagnetic field under a single set of equations, but then the task has become even more difficult, with the discovery of two other basic field: the weak interaction field and strong interaction field. Most physicists thought Einstein's quest was hopeless, and in fact he never succeeded. But Einstein was convinced such a basic harmony and simplicity existed in nature, he kept his chin up, went ahead along his own road. Because he was apart from the mainstream of physical research - quantum field theory, he was very lone in his old age, but he was fearless. He still prepared to keep on his mathematical calculation of unified field theory on his sickbed until the day before his death. He said with a sigh before his death: I cannot finish this work, it will be forgotten, but it will be rediscovered in the future. (quoted from http://xz.vip.sina.com/ebdl6.htm)
Einstein did manage to develop a theory which "wrapped" electromagnetism and gravitation into a common metric tensor. In one of his formulations of a unified field theory (called Einstein-Schrodinger Theory), gravitation was wrapped into the symmetric part of the metric tensor, while electromagnetism was wrapped into the antisymmetric part of the metric tensor. This wrapping is possible because electromagnetism and gravity share some mathematical similarities. They both have a stress-energy tensor. The electric charge is analogous to the gravitational mass. The magnetic moment is analogous to the angular momentum moment. The electric potential and electric field are analogous to the gravitational potential and gravitational field, respectively. Finally, the magnetic field is analogous to the magneto-gravitic field.
The mathematical wrapper which Einstein developed exploits this analogy. However, the analogy between electromagnetism and gravity breaks down at higher field strengths when nonlinear field effects set in. As a result, Einstein-Schroding theory correctly describes electromagnetism and gravity at low field strengths where they are not coupled to each other. However, it does not describe the interactions between electromagnetism and gravitation which occur at higher field strengths. Thus, Einstein-Schrodinger theory achieved an approximate mathematical unification, but no real physical unification of electromagnetism and gravity. In this sense, it did not really achieve its objective.
Kaluza and Klein developed an alternative wrapper for electromagnetism and gravitation. Instead of wrapping electromagnetism into the antisymmetric part of the metric tensor, they retained a symmetric metric tensor but added a fifth dimension. They were able to show that Maxwell's Laws and General Relativity can be expressed in terms of their five-dimensional metric tensor. Again, this exploits the analogies between electromagnetism and gravity.
The problem with Einstein's unified field theory and Kaluza-Klein's unified field theory is that they don't address the fundamental issue. They still treat gravitation and electromagnetism as two completely separate interactions. Neither theory can tell you how a gravitational field is fundamentally produced by a charged particle.
Today, the search for a unified field theory has been replaced by loftier goals. Physicists are now looking for a so-called Theory of Everything (TOE) which will unify not only electromagnetism and gravity, but also the nuclear interactions and other potential physical forces such as inflation and "dark energy". At the time of Einstein, modern particle physics had not yet been developed and the strong and weak nuclear interactions were not well understood.