String theory is the branch of science that investigates one of the most fundamental questions of physics and philosophy — how long is a piece of string? In spite of many years of theoretical and practical work, the question remains unanswered.
Due to the fact that string theory makes few predictions or falsifiable experiments, some doubt that it is a real theory at all, but just a hypothetical mathematical framework for particle physics. That said, it's promising enough to have all but monopolised funding for that level of theoretical study.
Superstring theory is the same thing but supersymmetric.
Some people argue that string theory addresses an entirely different question, but if you listen to them talk about it, you quickly realise that they may as well be talking about the origins of matter, energy and the universe, for all the sense they make. In fact they take the whole concept of talking gibberish off into uncounted dimensions, before they even start to try to measure the string's length.
One of the most famous theories of modern particle physics among the popular science aficionados, the various superstring theories, and the related D-brane theory, are attempts to explain the various subatomic particles and the particles which carry forces between them as the vibrations of tiny strings.
String theory aims to marry up the two worlds of quantum mechanics and relativity and explain all four fundamental forces: strong, weak, electromagnetic and gravity. The first three are well described by quantum theory (the theory that describes the very small, such as electrons and individual atoms); quantum electrodynamics addresses the electroweak interaction (a unification of electromagnetism and the weak interaction), and quantum chromodynamics addresses the strong interaction. Gravity is best described by general relativity, which describes the warping of space because of the presence of matter, producing the phenomenon of gravity.
Because quantum theory models all particles as specific points (i.e, without any size or volume associated with them), rather strange things happen when gravity is applied - the calculations don't work and the theory falls apart. Clearly, gravity exists, so must be explained by a complete theory of physics. More specifically, the uncertainty principle implies that, as you look at smaller and smaller chunks of spacetime, you will see more and more "virtual particles" bubbling-up out of the void. These virtual particles have mass-energy to them, and so warp the fabric of spacetime, to the point that it becomes "foamy" and can therefore no longer be described by general relativity (which assumes that spacetime is locally flat). String theory removes this problem by postulating that elementary particles are not infinitessimally-small "points," but strings of finite length, beneath which it is no longer meaningful to talk about physics at all. Thus, the arbitrary "foaminess" never comes into play.
Many theories exist which attempt to address both the electroweak interaction and the strong interaction (these theories are called "grand unified theories"), and one of them may be proved true when the Large Hadron Collider begins operating at full capacity. Fairly few theories exist which attempt to unify gravity with a Grand Unified Theory - superstring theory is such a framework.
 The general idea
Superstring theory postulates that at the scale of the Planck length, tiny strings vibrate in such a way that the vibrations give rise to the different properties we observe in particles - mass, charge, color, spin, etc. Five different superstring theories exist, each of which postulate a different number of dimensions, and different kinds of strings. It is possible that all five theories are special cases of a larger theory, called M-brane theory, or just M theory.
String theory addresses this issue by assuming that particles are not exactly "points", but tiny lengths, hence the term "string". These are hypothesized to extend in multiple dimensions (up to 11 as defined by "M-Theory" which itself is used to describe aspects of string theory) - not just in "strings" but in "membranes" - in addition to the x, y and z that can be macroscopically observed (hence it's difficult to get experimental evidence for them). The fact that they may extend in more than just the normal dimensions of space time is thought to be able to explain why gravity is so much weaker than the other forces, even though it operates on a much larger length scale. Beyond that, string theory involves some hair-raising mathematics, some deep questions about the nature of space and time, and also postulates a potential origin for the universe itself wherein two membranes collide in hyper-dimensional space. Or, it turns out to be bollocks and looped quantum gravity takes over the world.
String theory is short on falsifiable predictions. Nevertheless, it's considered promising enough to have all but taken over decently-funded theoretical high-energy physics. This is itself considered a problem by many (proper, non-crank) physicists who think thirty years is quite long enough for string theory to have come up with a verified falsifiable prediction and have problems getting funding for research that isn't string theory. These factors make String Theory a potential modern protoscience, although we will still have to wait to see if it develops into a fully grown and fully accepted scientific branch.
No, statistically no-one reading this article actually understands this beyond the "PHYSICIST make FIRE with STRING THEORY!" level.