There are 17 different symmetry groups under which any wallpaper pattern can be classified. For an excellent discussion of the 17 groups, click here (there are additional sites devoted to this topic which can be accessed via my Links page).
The notation works as follows: The first letter of the group name is p or c, depending on whether the lattice is primitive or centered. After that comes a digit which represents the highest order of rotation. The letter m represents a reflection, and g represents a glide reflection. So, for example, p4mm is a primmitive lattice with a 90o rotation, and reflections along two different axes.
The purpose of this section is not to explain about the groups, since that is so well done elsewhere, but to show how to create your own patterns (tessellating or not) using Microsoft Paint. Note that since Paint will only rotate an image by a multiple of 90o, it is impossible to create wallpapers in groups p3, p3lm, p3ml, p6, or p6mm using my technique, since the p3 groups require a 120o rotation and the p6 groups require a 60o rotation.
In all the sections that follow, I use the following notation:
x represents a drawn image in Paint.
Hx represents the image that results from selecting image x and doing a "Flip - Horizontal" on it.
Vx represents the image that results from selecting image x and doing a "Flip - Vertical" on it.
90x, 180x, 270x represent the images that result from rotating image x by 90, 180 or 270 degrees.
For function composition, the order is right to left. For example, H90x means first rotate image x by 90o and then do a "Flip - Horizontal" on it.
When designing, I always draw my lattice in first, in grey. The lattice will be made up of squares, rectangles, or right isosceles triangles, depending on the group. (Obviously, a group that permits a rectangular lattice will also permit a square lattice, but not necessarily the other way around. I generally use a square for both.) Once the design is finished (in black & white), I select the lattice and copy it to a separate file. Then I open that file, remove all the grey lines, and color it in.
p1 (rectangular/square)
p2 (rectangular/square)
p4 (square) ** Coming soon **
pm (rectangular/square) ** Coming soon **
pg (rectangular/square) ** Coming soon **
pmm (rectangular/square) ** Coming soon **
pgg (rectangular/square) ** Coming soon **
pmg (rectangular/square) ** Coming soon **
p4m (triangular) ** Coming soon **
p4g (square) ** Coming soon **
cm (rectangular/square) ** Coming soon **
cmm (triangular) ** Coming soon **