"Alebrije" is a made up word meaning "nightmare."  Pedro Linares woke up from a coma and started doing these creatures that he had seen while "asleep" in paper mache. It was he who originated the word. These whimsical "nightmares" are now done by artists all over Mexico in many mediums, but paper mache is the way they started.  The word has also come to refer to Oaxacan Woodcarvings, also known as "animalitos."
It is said that, if you keep one of these creatures in your bedroom, when a nightmare comes along, it will see yours, say, "Oh, this person already has one," and leave.