Wallpaper comes in an infinite variety of patterns, from repeating peacocks to intertwined flowers to Escher creatures to simple stripes. Mathematically, though, there are only a finite number of distinct types of wallpaper. It turns out that all the elaborate patterns of the world's walls can be stripped down to their bare bones, analyzed for which symmetries they display, and identified as one or another of exactly 17 so-called wallpaper groups. (Yes, 17 - it appears again.)
Wallpaper is only the historical point of reference for a more general statement about all two-dimensional repeating patterns, including things like tessellations, honeycomb, checkerboards, and chain-link fences too. Long before mathematicians rigorously classified planar patterns as "p4g", "pmm", or any of the other wallpaper groups, the ancient Egyptians discovered and plastered all 17 of them up on the walls of their rooms and tombs.
The classification of the wallpaper groups is based on how individual segments of a pattern, called unit cells, fit together. To determine how they fit, and which group they fit into, you test how you can transform the pattern and still end up with it looking how it did before. You test whether you can translate it (by shifting the unit cells over one place and ending up with the same thing), rotate it, reflect the pattern across a line, or "glide reflect" it, which means reflecting it across a line and simultaneously shifting it.
Based on those four types of symmetries, and which of them a given pattern possesses, it can be categorized in one of the 17 groups. Sometimes it is obvious that two wallpaper patterns are members of the same group, as in this case:
(Group cm - reflections and at least one glide reflection along an axis other than the reflection axes)
And sometimes patterns within the same group look very different:
(Group pmg - two centers of 180 degree rotation, a reflection axis, perpendicular glide reflection axes)
The mathematical proof of there being exactly 17 groups gets rather hairy, but fortunately someone wonderful has created an Applet allowing you to study the wallpaper groups directly: by designing wallpaper patterns of your own in mere seconds. It's fun!