What does string theory prove? How possible is string theory?
During the course of the 20^{th} century, two basic theories of physics have emerged. Historically, the first is the theory of relativity, developed mainly by Einstein. There are really two theories of relativity: special relativity was introduced in 1905. This theory deals with the motion of particles at very high speeds (close to the speed of light). It relies on the experimental evidence that the speed of light is independent on the speed of the observer, as demonstrated by the famous experiment conducted by Michelson and Morley. In 1915, Einstein introduced the general theory of relativity, which replaces Newton's theory of gravity. Einstein's motivation was to put gravity in the framework of special relativity, namely to ensure that any gravitational signal does not travel faster than light. He managed to do so by explaining gravity as a geometrical effect - the existence of mass stretches the time and space in such a way that bodies that move nearby feel that they "fall" into the body.
A few years later, in the 1920's, the second triumph of 20^{th} century physics had emerged - the theory of quantum mechanics, by people such as de-Broglie, Planck, Schrodinger, Heisenberg and others. A series of experiments conducted in the late 19^{th} and early 20^{th} century proved that on a very small - atomic scale, particles and light do not obey the classical laws of motion developed by Newton in the 17^{th} century. A new set of physical laws had to be developed in order to accurately describe the behavior of particles, as well as light, on a very small scale. As it turns out, the laws of physics on that scale are very different than the laws of physics on bigger, everyday scale. For example, on these scales, particles show wave-like behavior, such as interference and diffraction that are not seen on bigger scale, and clearly were not predicted by the laws of classical mechanics.
These two theories- relativity and quantum mechanics were derived independently, and rely on different experimental facts. In fact, as became clear during the second half of the 1920's, these two theories are inconsistent with each other. For example, in certain cases the predictions of quantum mechanics require particles to travel faster than light, which is impossible in relativity. Already in the mid 1920's, scientists had noted this problem, and began a major effort in an attempt to unify these two theories into one, fully self-consistent theory that will properly describe the laws of nature in both the very small as well as the very large scales. Initially, quantum mechanics was merged with special relativity. This marriage gave birth to the theory known as quantum field theory, initially suggested in the 1930's, and further developed all along the 20^{th} century by people such as Dirac, Feynman, Schwinger, Wilson and others. This unified theory turned out to be extremely successful: it gave several very important predictions, such as the existence of anti-matter, which were later on discovered and proved this theory to be correct. In fact, this theory is widely accepted today as the best known basic theory of nature. It forms the basis of modern particle physics - what is known as "the standard model" of particles physics, that explains the basic ingredients of matter (quarks, leptons, etc.) and the interactions between them.
All along the 20^{th} century and until today, there are numerous attempts to unify, in a similar way, the theory of quantum mechanics with the theory of general (rather than special) relativity. The idea - or maybe "dream" is a more appropriate phrasing here, is to unify these two theories into one, fully self-consistent theory that will explain all the basic laws of nature. Despite great efforts, until today, no such complete "theory of everything" is fully established. The name of the modern version of this theory, still being constructed from the 1960's until today, is string theory (today there is a further development called "M-theory").
The basic idea of string theory is that all the laws of nature originate from one "fundamental" law, which manifests itself in different ways under different conditions, thereby seen to us as separate "basic" laws of physics. Whether this is the case or not, nobody knows. Physics is an experimental science; a theory is accepted only if it can explain the results of a scientific experiment, or observation. As opposed to all other physical theories that were developed in history, currently there is no experiment that shows any need for modifying the known laws of nature; and, in fact, nobody expects any such experiment to be conducted in the near future. According to string theory, the "fundamental" law should be manifested only close to, or at energy which is 15 orders of magnitude (namely, million-billion times) higher than the maximum energy that was ever achieved by mankind (at the large hadron collider in CERN). Thus, it will likely never be achieved during our lifetime.
Thus, in a very fundamental sense, string theory is a speculative science. There is no experimental evidence that string theory is the correct description of our world and scant hope that hard evidence for it will arise in the near future. Moreover, string theory is very much a work in progress and certain aspects of the theory are far from being understood. Unresolved issues abound and it seems likely that the final formulation has yet to be written.
Still, many people like it for various reasons. First, it is great to imagine the idea that indeed there exists only a single fundamental law, from which all the laws of nature are derived. Second, many people are puzzled by the fact that quantum mechanics and general relativity are so fundamentally different. People like Hawking consider boundary cases, in which both quantum mechanical and general relativistic effects should co-exist (such as in the famous "Hawking radiation", radiation that originates from quantum fluctuations very close to a black hole, and should emerge from black holes), and try to construct a self-consistent way of explaining all of that. Third, theoretical developments so far seem promising: it was proven that under the appropriate conditions, string theory does give rise to known laws of nature, such as quantum field theory (at least partially).
Fourth, while string theory cannot at present offer falsifiable predictions, it has nonetheless inspired new and imaginative proposals for solving outstanding problems in particle physics and cosmology. Finally, study of string theory gives new ideas and techniques that are implemented not only in other branches of physics, but in mathematics as well, and suggest new directions and insights in mathematics.