Superstring theory is an attempt to explain all of the particles and fundamental forces of nature in one theory by modeling them as vibrations of tiny supersymmetric strings. It is considered one of the most promising candidate theories of quantum gravity.
At present, the deepest problem in theoretical physics is harmonizing the theory of general relativity, which describes gravitation and applies to large-scale structures (stars, planets, galaxies), with quantum mechanics which describes the other three fundamental forces acting on the microscopic scale.
The development of a quantum field theory of a force invariably results in infinite (and therefore useless) answers. Physicists have developed mathematical techniques (renormalization) to eliminate these infinities which work for the electromagnetic, strong nuclear and weak nuclear forces, but not gravity. Thus the development of a quantum theory of gravity must come about by different means than were used for the other forces.
The basic idea is that the fundamental constituents of reality are strings of the Planck length (about 10^{-33} cm) which vibrate at resonant frequencies. The graviton (the proposed messenger particle of the gravitational force), for example, is predicted by the theory to be a string with wave amplitude zero. Another key insight provided by the theory is that no measurable differences can be detected between strings that wrap around dimensions smaller than themselves and those that move along larger dimensions (i.e., effects in a dimension of size R equal those whose size is 1/R). Singularities are avoided because the observed consequences of "big crunches" never reach zero size.
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Our physical space is observed to have only four large dimensions, and a physical theory must take this into account. But nothing prevents one from having more than 4 dimensions per se. In the case of string theory, consistency requires spacetime to have 10, 11 or 26 dimensions. The conflict between observation and theory is resolved by making the unobserved dimensions compact dimensions.
Our minds have a hard time visualizing higher dimensions because we can only move in three spatial dimensions. And even then, we only see in 2+1 dimensions; vision in 3 dimensions would allow one to see all sides of an object simultaneously. One way of dealing with this limitation is to not try to visualize higher dimensions at all but to just think of them as extra numbers in the equations that describe the way the world works. This opens the question of whether these 'extra numbers' can be investigated directly in any experiment (that must show, ultimately, different results in 1, 2, or 2+1 dimensions to a human scientist). This, in turn, opens the question whether models that rely on such abstract modelling (and potentially impossibly huge experimental apparatus) can be considered 'scientific'.
Superstring theory is not the first theory to propose extra spatial dimensions; see Kaluza-Klein theory. Modern string theory relies on the mathematics of folds, knots, and topology, which was largely developed after Kaluza & Klein, and has made physical theories relying on extra dimensions much more credible.
Theoretical physicists were troubled by the existence of five separate superstring theories. This has been solved by the Second Superstring Revolution in the 1990s during which the five superstring theories were discovered to be different limits of a single underlying theory. See M-theory.
The five consistent superstring theories are: type I, type IIA, type IIB, Heterotic E8 X E8 ( or HEt), and Heterotic SO(32) (or HOt).
Heterotic SO(32) is slightly inaccurate since among the SO(32) Lie groups, string theory singles out Spin(32)/Z2.