The fourth dimension is generally understood to refer to a hypothetical fourth spatial dimension, added on to the standard three dimensions. It should not be confused with the view of space-time, which adds a fourth dimension of time to the universe. The space in which this dimension exists is referred to as 4-dimensional Euclidean space.
Beginning in the early part of the 19th century, people began to consider the possibilities of a fourth dimension of space. Mobius, for example, understood that, in this dimension, a three dimensional object could be taken and rotated on to its mirror image. The most common form of this, the four dimensional cube or tesseract, is generally used as a visual representation of it. Later in the century, Riemann set out the foundations for true four-dimensional geometry, which later mathematicians would build on.
In the three dimensional world, people can look at all space as existing on three planes. All things can move along three different axes: altitude, latitude, and longitude. Altitude would cover the up and down movements, latitude the north and south or forward and backward movements, and longitude the east and west or left and right movements. Each pair of directions is at a right angle to the others, and therefore is referred to as mutually orthogonal.
In the fourth dimension, these same three axes continue to exist. Added to them, however, is another axis entirely. While the three common axes are generally referred to as the x, y, and z axes, the fourth falls on the w axis. The directions that objects move along in that dimension are generally called ana and kata. These terms were coined by Charles Hinton, a British mathematician and sci-fi author, who was particularly interested in the idea. He also coined the term "tesseract" to describe the four dimensional cube.
Understanding the fourth dimension in practical terms can be rather difficult. After all, if someone is told to move five steps forward, six steps to the left, and two steps up, she would know how to move, and where she would end up in relation to where she began. If, on the other hand, a person was told to also move nine steps ana, or five steps kata, she would have no concrete way to understand that, or to visualize where it would place her.
There is a good tool to understand how to visualize this dimension, however, and that is by first looking at how the third dimension is drawn. After all, a piece of paper is a two-dimension object, roughly, and so cannot truly convey a three dimensional object, like a cube. Nonetheless, drawing a cube, and representing three-dimensional space in two dimensions, turns out to be surprisingly easy. What one does is to simply draw two sets of two-dimensional cubes, or squares, and then connect them with diagonal lines that link the vertices. To draw a tesseract, or hypercube, one can follow a similar procedure, drawing multiple cubes and connecting their vertices as well.