Chaim Goodman-Strauss's site has explanations of symmetries in the plane, with pictures
http://comp.uark.edu/~cgstraus/symmetry.unit/
Kali is a program that generates symmetric patterns. The Mac and Windows versions can be found at
http://humber.northnet.org/weeks/index/Kali.html
A Java version of Kali is an interactive online version at
http://thinks.com/java/kali/kali.htm
Note: In addition to Kali, Jeff Weeks has some other interesting math software at