Our intuition and observation tell us that there are three spatial dimensions - the "left-right" dimension, the "up-down" dimension, and the "back-forth" dimension. In out experience, any motion we undergo is motion through some combination of these three dimensions, and any location in the universe can be specified with three coordinates corresponding to these dimensions. Einstein's work shows that time is equivalent to these three dimensions - it can be considered the "future-past" dimension. This gives added specificity - events are designated by three coordinates of location and one of time. These are the four dimensions, three space and one time, that are used in relativity and quantum mechanics in the form of spacetime, the "fabric" of the universe.
In 1919, the obscure Polish mathematician Theodor Kaluza of the University of Königsberg suggested that the universe might somehow have more than three spatial dimensions. He sent a paper to Einstein revealing his suggestion and showing how extra dimensions provide a way to combine general relativity and Maxwell's electromagnetic theory by stating that light waves are actually vibrations in a fifth dimension. However, Einstein waffled in his enthusiasm for the idea, at first responding with interest, then with skepticism, discouraging Kaluza from publishing his paper. Then, two years later, he apologized to Kaluza and offered to publish the paper. Subsequently, the theory was refined by the Swedish mathematician Oskar Klein, whose 1926 revision added ideas from the emerging field of quantum mechanics. His calculations made explicit the idea that extra dimensions may be "curled up" so that they are hidden from view. Additionally, he predicted the size of such a dimension: about the Planck length. The theory became known as Kaluza-Klein theory. However, later experiments showed that the theory's predictions were in conflict with experimental data. Physicists of the time were much more interested in probing the microscopic world with their new quantum theory than they were in obscure theories returning incorrect predictions, so Kaluza-Klein theory was abandoned.
Kaluza-Klein theory postulates the existence of extra curled-up dimensions beyond the four with which we are familiar. But what exactly is a curled-up dimension, and what does it look like? A popular example, given by Brian Greene in The Elegant Universe, is that of life on a garden hose with such a tiny diameter that this dimension is unnoticeable; its inhabitants live exclusively on its surface. 
This life takes place in one spatial dimension, the "back-forth" dimension mentioned above. Everything in this odd universe appears to its inhabitants as a point, since this is all that can be seen in one dimension. Furthermore, a "being" in this universe would be a line locked into position by the surrounding other beings - there is no room to move "past" anything, because the entire universe is a line. In effect, these beings' bodies fill up their universe (see diagram).
If, in this scenario, the curled-up dimension were to expand, the linelike beings could move about in two dimensions - back-forth and left-right. These beings have room to become two-dimensional - in the image, they are bloblike, but they could just as easily be regular plane figures, as they are represented in Edwin Abbott's classic Flatland.
Although Kaluza's revolutionary idea had promise, it was abandoned in the wake of the quantum-mechanics craze that swept physics in the 1920s. However, by the early 1970s the standard model was in place and most physicists felt as though everything important about the nongravitational forces had been discovered. After numerous setbacks regarding the merger of quantum mechanics and general relativity, physicists became receptive to unusual ideas - including the resurrection of Kaluza-Klein theory. It was suggested that Kaluza's one additional spatial dimension was not enough - additional extra dimensions were needed to fully unify relativity and electromagnetism. Theories called higher-dimensional supergravity theories sprang up, partially incorporating gravity, supersymmetry, and higher dimensions. These helped to calm the quantum fluctuations barring a sensible theory of gravity, but did not do enough to eliminate the problematic infinities associated with the attempt.
Within quantum mechanics, predictive power is reduced to probabilities, which of course are numbers between 0 and 1. Although string theory eliminated the infinities associated with quantum mechanical descriptions of gravity, it also sometimes yielded negative probabilities in certain quantum-mechanical calculations. The reason behind this became evident: the erroneous calculations involved quantities sensitive to the number of dimensions in which a string can vibrate. Theorists had been carrying out the calculations in three spatial dimensions, but this proved insufficient - the negative probabilities canceled out if a string could vibrate in nine spatial dimensions. This at first seemed the end of string theory - reality seemed to impose the three-dimensional restriction. However, Kaluza-Klein theory provided a way out. In the Kaluza-Klein formulation of string theory, there are three extended spatial dimensions, one time dimension, and six curled-up dimensions. (As will be revisited in future parts of the series, it was later shown that the nine dimensions are approximations - in reality, there are seven curled-up dimensions, yielding a total of eleven.)
Strings vibrate through all the spatial dimensions, meaning that their precise vibrational pattern is affected by the shape of the curled-up dimensions as well as the extended ones. It follows from this statement that the shape of the curled-up dimensions has an indirect effect on the observed elementary particles, since these are merely reflections of different vibrational patterns. This topic is discussed more fully in the next page in the series.
Created by Dan Corbett, Kate Stafford, and Patrick Wright for ThinkQuest.