A tesseract is a hypercube, the four-dimensional equivalent of a cube, which has been unfolded into three dimensions.

Since four-dimensional space is difficult for the human mind to visualize, the best way to think of this is to remove a dimension. Every school child has made a cube from a two-dimensional piece of paper by cutting out the following shape:

       _____
      |     |
     A|     |
 _A___|_____|___________
|     |     |     |     |
|     |     |     |     |
|___ _|_____|__ __|__ __|
      |     |
      |     |
      |_____|

and folded along the lines to create a three-dimensional cube. This is called the net of the cube. When we fold it up, the edges labelled A, become the same edge in the cube. Similarly, other pairs of edges merge to form one edge in the cube

The three-dimensional 'net' of the tesseract looks like this:

    /\  /\  /\
   /  \/  \/  \
  /   /    \   \
 |\  |\    /|  /|
 | \ | \  / | / |
 |  \|  \/  |/  |
  \  /\  | /\  /
   \/  \ |/  \/
   /    \/    \
  |\    /\    /|
  | \  / |\  / |
  |  \/  | \/  |
   \ |  /|\ |  /
    \| / |/\| /
     \/ \/  \/
      \  | /
       \ |/
        \/

/me realises that representing four-dimensional geometry in ASCII art is probably a mistake

In case that isn't clear, it is a stack of 4 cubes, with four more cubes arranged in a cross around the second cube from the top. When we 'fold' this up, the top face of the cube at the top of the stack merges with the bottom face of the bottom cube, the adjacent edges of the cubes in the cross join, and so on to form the hypercube.

As a interesting historical note, in the picture Christus Hypercubus, Salvador Dali depicted Christ being crucified on a tesseract.

The hypercube in R^4 (four-space), also called the 8-cell, is known as a tesseract. It has the Schläfli symbol {4,3,3}, and vertices (+-1,+-1,+-1,+-1). A tesseract has 16 vertices, 32 edges, 24 squares, and 8 cubes.

Also, a concept used in the popular Sci-Fi children's book by Madeleine L'Engle A Wrinkle in Time. The tesseract is a concept used to explain time travel in the sense that a line is the shortest way to travel between two points

O--------O

However, if you can imagine placing your finger in the middle of that line, and folding the line so that the two endpoints are side by side, then you can see how the line is technically not the shortest way, but rather to fold it. The book used the example of an ant crossing this distance as it would travel between two points on the hem of a skirt.

Also used in the short story '--And He Built a Crooked House' by Robert Heinlein. An architect built an 'unfolded' hypercube as a house, similar to Tristan's drawing above, but upside down. The first cube on the bottom was the garage and utilities, the second floor (TWIAVBP UK: First floor) was the living room and such with a cube on each side, then another two above. This was built in California, and during an earthquake, the unstable unfolded four-dimensional structure collapsed into its more stable form...leaving the people inside slightly trapped. Each exit had two ways to go through it; the way that would have led to another room of the house and if you were either distracted or had the right mindset when going through, you would go outside of the cube. One of them discovered this by accident, by falling out a window into a rosebush.