String Theory
This article depends on
many others, fortunately most of which are already written up elsewhere. Out of necessity, I have sometimes put in brief descriptions of supporting theories, but to get a better understanding please follow the links to the relevant write ups.
Theories that string theory builds on
 General Relativity A particle is specified by a fourvector X(t) for a coordinate t running along the particle's world line. The distance along this line is determined by the metric tensor (which depends on the matterenergy content of the Universe)  the theory states that a freely falling particle moves in such a manner as to minimise the length of the world line (it is thus called a geodesic).
 Quantum Field Theory A classical field theory (eg. Relativistic Electrodynamics) involving a classical field A(x) defined at all spacetime points x is promoted to a quantum field by promoting A(x) from being just a number to being an operator. This is done so that the operator field creates a particle state from a hypothetical vacuum state 0> at some point in spacetime; we write the state as x> = A(x)0>. Here, A(x) is the vector potential for the electromagnetic field, and the particles created by it in this sense are called photons. Quantum field theories depend on being able to create and destroy particle as various points, which is necessary for the theory to be able to cope with interactions between different types of particle.
Interactions occur between different quantum field in a manner prosrcibed by Feynman diagrams. Quantum Electrodynamics (QED) provides a good example. in this theory, there is both the electromagnetic field A (as above), and the electron field Y (read this a greek phi). When considered seperately (not interacting), Maxwell's equations govern the motion of the photons (A) and the Dirac equation governs the electrons (Y). When interacting, the world lines meet at a point:

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At this point the Y field destroys an electron state, and A creates a photon while Y creates another electron.
Both these previous theories are based on the idea of point particles. String theory starts with the bold statement that instead particle should be thought of as extended objects  which at any given moment in time appear to be one dimensional strings as opposed to zero dimensional point objects. That we have not yet observed particle to be this way is accounted by the lengths of the strings being so small as to be indistinguishable from merely being point objects. Formally, the string is specified by X(s,t)  rather than just X(t)  where s is a new coordinate labelling the internal spatial degree of freedom of the string. When you draw out the function X(s,t) it produces a two dimensional
surface (that's a tautology, I know) called the particle's
world sheet.
There are two essentially different configurations for the string: those for which the two ends meet, called
closed strings and giving tubelike world sheets, and those that do not (ie. the ends are free), called
open strings and giving bandlike world sheets. I will return to the distinction between the two later on.
How does this change the theory?
To make further progress, let's see how string theory inherits the characteristics of the two existing theories:
 Relativity: Previously, with the world line X(t), minimising its length gave the geodesic required for particle motion. Here, the natural generalisation is to minimise the area of the world sheet. This provides us with the (classical) equation of motion for the string:
(d^{2}/ds^{2})X(s,t) = (d^{2}/dt^{2})X(s,t)
This has a general solution
where L and R can be any two functions (respectively called left and right moving). When you impose the string boundary conditions, both these functions must be periodic, and for an open string they must be identical. This gives a large amount of variability to the solution, which is best understood by decomposing L and R as Fourier series  as a superposition of different resonances of the string.
 QFT:
As with going from any classical field theory to a quantum one, the first step is to promote the function X(s,t) to being an operator, with the understanding that the quantum field X now creates an destroys particles from a hypothetical vacuum 0>. Because of this, the different fourier modes (resonances) due to the above world sheet equation of motion become operators also, and are interpreted as generating different types of particle. With further examination, the species of particle derived in this way coincide beautifully with the known the standard model particles (electrons, quarks, gluons etc.), and also accomodate for things like gravitons which are just what you need for a quantum theory of gravity.
Also, having world sheets instead of world lines pleasantly clears up the picture for interactions in QFT: if you start with the usual Feynman diagram of two electrons and a photon joining at a point from QED and redraw it as a picture for strings, you instead have something that looks like a pair of trousers.
.. ++;
=##==###, ##. ,##=
##= #+ #. =#
#= +#= #X .#
# ++# ## #
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What is nicer about the string diagram is that there is no longer any real distinction between what happens when particles are simply propagating through spacetime (the lines in the QED picture) and when they are interacting (the verticies of the QED picture)  because when you look closely enough at the pairoftrousers diagram, any one region looks just like any other. The interpretation of this is that interactions are a natural feature of any string theory, that is, interactions emerge from the theory without any extra work (unlike QFTs).
String interactions
Given that strings can be either open or closed, there are a variety of possible interactions.
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+# X# #x x# #, #x #x #+
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X#;;, x# x##=. .x#+
,;;==; ,=xxx+=.
(here is an openopenopen interaction, and a closedopenopen interaction)
A theory featuring only closed strings is entirely consistent, but one which has open strings must also include closed ones, as the above interactions allow the two ends to meet and form a closed string. This gives a variety of possible string theories, a good summary of which can be found under
CapnTrippy's
superstring theory wrtieup.
Higher dimensions
Given a string world sheet, there are many different ways we can parameterise the surface as X(s,t) which all really describe the same stringlike particle's history, and hence describe exactly the same physics. What we have is a
redundancy in our mathematical description of the string  imagine this as like being presented by a psychologist with five different coloured pens and being asked to write your name down on a piece of paper. It doesn't matter which colour you use, but you have to use one.
This causes a potential problem that the states we create (in a QFTheoreticsense) with the string have this redundancy too, and we need a way to make them indistinguishable. Continuing the analogy, you don't want a psychologist coming along and drawing any kind of conclusion based on the colour of pen you used, when it is only your name that matters. This is done by imposing a
symmetry between the equivalent states, and claiming that any two states that can be related by this symmetry describe the same physical particle, and is called the conformal symmetry of the string. The analogy here would be that you need to make sure the psychologist is colourblind (that there is a symmetry relating the five colours).
Sounding very obscure as this does, the important thing about this digression is that the matter of imposing this symmetry on the string states is not without its mathematical complications, and can only be done consistently if the number of dimensions of spacetime is equal to 26. Oh. Being used to a four dimensional universe, this would appear to be the doom of string theory. However, string theortists have developed a good way of sweeping this problem under the rug by introducing the notion of compactification.
The classic way to explain this is by picturing a garden hose  far away it appears lineline, so thin as to be assumed one dimensional; yet as we know, when examined close up the hose is in fact a long cylinder which is two dimensional. One extra dimension, corresponding looping around the cylinder has been supressed (not coincidentally, in the same way strings themsleves could be mistaken for point particles), and this can happen because that circle has the topological property of being compact (a compact object has a property of having a finite size, and thus possibly a very small size in this case).
The claim is then that the extra 22 dimensions are in this sense compact, and so small that they have not yet been noticed. Of course, there is much variety as to what form they could take, as there are a large variety of compact shapes (eg. any combination of circles, tori, spheres etc.). The nice thing about this is that these different 'shapes' for the extra dimensions correspond to different possible standard models of particle interactions for the theory  so the incumberence of extra dimenions then becomes a virtue! In this interpretation, the way particles interact is a consequence of the small scale geometry of the Universe. Oooh.
Superstrings
I lied when I mentioned that electrons and quarks feature in the above theory; I was making a temporary attempt to hide a complication to the theory. In fact, the above theory is only any good for describing
bosons (such as the photon and the graviton  which are both associated with
forces). Electrons and quarks, the consitiuents of
matter, fall into the other category of particles called
fermions.
A theory with bosons alone is entirely consitent (provided there are 26 spacetime dimensions), yet a theory incorporating both species of particles is what is required. In this case, we need not only conformaly symmetry for the world sheet, but also world sheet
supersymmetry. The two combined form a larger symmetry called superconformal symmetry. Supersymmetry (SUSY) has already got its own node, so please read that if you really want to know what it is. In short SUSY is:
 A symmetry which exchanges bosons and fermions
 AND a symmetry of spacetime!
Quite how it can be both is really deep and amazing due property of both the universe we inhabit and the matter within it that is only really explained by
Mtheory. Putting aside the philosophical musings as to what SUSY really means, what matters for superstring theory is that the theory must now contain both bosons and fermions, and physical states are now related by the superconformal symmetry. Imposing the superconformal symmetry requires
10 and not 26 dimensions for the consistency of the theory.