Alebrijes (ah-lay-bree-hays) began as papier mache fantasy animals. They were originally created by Pedro Linares after a dream he had while sick, in order to illustrate for his family the creatures he had seen in his dreams. In Oaxaca and the surrounding area, this art form has been recreated using wood carvings.
Using copal wood, the artist carefully carves an animal figure. At this stage, the alebrije looks like this.

The alebrije is then painted according to the whim of the artist, using brightly colored acrylic paints.
Each artisan family has their own style, and many times you can tell where an alebrije originated just by looking at the type of carving style or the design used for the intricate paints which are applied.
No two alebrijes will be exactly the same, though sometimes you may find some that are similar.