As I understand it, a MOIRE pattern is an optical pattern created when two arrays are superimposed on top of eachother-like two layers of sceening. When you view this, an interesting pattern is created.
What are these patterns used for? 
In my experience, moire patterns are the opposite of useful–you try to avoid them at all costs. You can get them when you scan newsprint, improperly downsample images with fine patterns, see them on TVs when a fine pattern appears, etc.
Wikipedia suggests one area where moire patterns are useful: microscopic strain measurement.
Moire patterns can be used for measuring: http://www.optics.arizona.edu/jcwyant/Optics513/ChapterNotes/Chapter04/MoireChapterSecured.pdf
Most times they are just an annoyance. Especially for owners of digital cameras.
Moire patterns are useful for illustrating and measuring surface topography (they use them for measuring scoliosis, for instance). The technique of Moire Deflectometry is useful for measuring optical properties – you can use it to determine radius of curvature, optical power, Modulation Transfer Function, and, in a different and more precise way, surface topography. The Rotlex corporation sells complex and full automated devices for measuring properties of eyeglasses using this technique.
Moire patterns are used by a lot of artists. I’ve seen some very clever work in thgis line. Dover books also sells several cute books filled with old 19th century engravings that seem o “move” when you slide a regular grating over it.
Finally, the patterns are interesting in their own right. Dover, again,m sells a book of various patterns printed on paper and on transparencies, and you can slide one over the other. The fascination with moire patterns apparently was a big thing in the druggie sixties, and I’ve seen films of concerts with projected sliding moire patterns behind the band. One of the most popular black light posters had the moire pattern of two slightly disarranged concentric circles.