Tesseract

A tesseract is a 4-dimensional cube. One way of putting it in perspective is in relation to similar forms in other dimensions.

All of the following figures are composed of sides of equal lengths that attach at right angles.

2 Dimensions

A square is a closed figure in 2-dimensions. It has 4 lines that meet each other at right angles with 2 lines attaching at each vertex. It has 1 “side.”

3 Dimensions

A cube is a closed figure in 3-dimensions. It has 12 lines (edges) that meet each other at right angles with 3 lines attaching at each of its 8 vertices. A cube has 6 “faces” or sides. Each face of a cube is a square. 

4 Dimensions

A tesseract, also known as a hypercube, is a closed figure in 4 dimensions. It has 32 edges that attach at right angles with 4 lines attaching at each of its 16 vertices.  It has 24 “faces” that make up 8 cubes, and, because we’re dealing with 4 dimensions, each of the “faces of a tesseract is actually a cube and there are 8 of them.

The best person to explain more about the tesseract is the Mathologer.