The original string theory, Bosonic String theory, was formulated in 26 dimensions, and addressed only bosons, not fermions.  It suggested that, rather than dimensionless points, bosons comprise tiny vibrating strings.  A major drawback was that it included tachyons, hypothetical particles that always traveled faster than light speed.  Superstring theory is an extension that includes fermions and supersymmetry, and dispenses with tachyons.  All particles comprise tiny vibrating "superstrings" at around the Planck length (about 10-33 cm).  They have resonant frequencies and harmonics that determine the particle represented by the string.  The theory also demands six additional spatial dimensions over the  four (three of space and one of time) that we currently observe.  There are various ways to define these dimensions mathematically so that they are present, but not observable directly; for  example, a six-dimensional Calabi-Yau space.  This means that at every point in space there is a Calabi-Yau "manifold" containing six dimensions, each of which is on the order of the  Planck length.  Unfortunately, there are a huge number of different forms that this manifold can take, and a major challenge is developing the mathematics to determine which space is  correct.

Physics

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### Quantum Gravity

There are several issues with Superstring & "M" theories.  Firstly, they have to be analyzed in a pre-existing space-time background until the mathematics is found to permit strings to generate their own background space-time.  Hopefully, when this happens, the space will have one time dimension and three expanded space dimensions!  But having to investigate strings that are not background independent is in direct conflict with General Relativity.  Another issue is that the theory intimates that it is not possible to probe below the Planck scale, and that, essentially, it does not exist.  However, there is a complementarity between very small and very large radii within the theories such that as something contracts and hits the Planck length, any further reduction in radius is the same as increasing the radius, avoiding becoming smaller.  This is known as "T" duality, and is a function of the compactified dimensions, and the fact that a string can "wind" itself around the circular dimension.  The number of times it winds itself is called the Winding Number.

A widely used mathematical tool in particle physics is perturbation theory.  Unfortunately, when the coupling constant in Superstring theory becomes larger than unity. The correction terms to perturbation get larger not smaller, so the technique becomes useless.  However, it appears that large values of the coupling constant in one versions of Superstring theory equates to low values of the coupling constant in another version, where perturbation is usable.  Type I theory at low energies (weakly coupled) is equivalent to Heterotic O (SO32) at high energies (strongly coupled).  This is called "S" duality, and has permitted research into strong coupling where previously, only very limited mathematical tools had been available.
In particle physics, very small distances correlate to very large energies.  Considering fundamental particles as dimensionless points means that they can approach infinitely close together resulting in infinitely large energies.  Strings are one dimensional, so are essentially "smeared out" so the forces do not become infinite, and a coherent description of both the very small and very large becomes possible.  All five Superstring theories naturally include the massless, spin-2 graviton and its super-partner the spin-3/2 gravitino.
Edward Witten proposed M-Theory in 11 dimensions, one more than Superstring theory, as a sort of umbrella theory that includes all five Superstring theories and Supergravity; here is a diagrammatic representation of M-Theory.  It includes "branes" that are multi-dimensional extensions of Superstrings that came out of M-Theory.  Essentially, a zero-brane is a point particle, a 1-brane is a Superstring, a 2-brane is a membrane etc.  It is theorized that we live in a 3-brane (three spatial dimensions)  that is embedded in a higher dimensional space.  The extra, eleventh, dimension may or may not be compactified like the six extra dimensions postulated in Superstring theory.  The Randall-Sundrum model, discussed in the next section on 5-Dimensional Warped Geometry Theory, is a somewhat simplified version that considers the three-space plus one time dimension we perceive, plus one extra dimension that is not compactified.
We have already indicated that there are two types of string; open ended and closed loops.  In the Brane World scenarios, the ends of open strings are tied to the branes.  Those that end on "our" 3-brane manifest themselves as the familiar fundamental particles and forces.  Not all strings are fixed to the brane.  Gravity is usually represented by closed loops that are not restricted to the brane but move freely in the bulk; that is in the extra dimension.
The Official String Theory site has a basic, non-mathematical Introduction to Superstrings, as well as a more advanced option for most topics.

Here is a Cambridge University article on String Theory and M-Theory.

Here is a fairly basic discussion on supertsrings, M-theory and brane world models