Really What Are They?
Epiphyllums are cacti which grow in the rainforest. These cacti are "epiphytes", or plants that grow in humus pockets of trees. Other examples of epiphytic plants include orchids and bromeliads. Epiphyllums and related species range from Mexico and the Caribbean through Central and South America. Although they are cacti, epiphyllums don't grow on the ground, and often lack the numerous spines so commonly found on desert cacti.
The name “Epiphyllum” means literally “upon the leaf” referring to how the flowers emerge from the strap-like branches. However epiphyllums do not actually have any leaves -- the flowers form directly on the branches of the plant. Epiphyllum hybrids are commonly referred to as "Orchid Cacti" because of their luminous blossoms, reminiscent of tropical orchids.
Most of what you see in the display are hybrids or cultivars. The ancestors of these hybrids come from the tropical American rainforest. There are both night and day blooming jungle cacti. Perhaps you or a family member had a “Night Blooming Cereus” or a “Queen of the Night” -- common names for a night blooming cactus which might be the species Epiphyllum oxypetalum. Other epiphytic cacti found in the jungles are Disocactus, Hylocereus (dragon fruit), Lepismium, Pfeiffera, Rhipsalis, Schlumbergera (Thanksgiving/Christmas cactus), Selenicereus, and others.
You might wonder how hybrid are named. These names are given upon registration of a hybrid. There is no protocol for names given to epiphyllum except there can be no duplicate names given and even that is only 99.99% the case.
How does hybridization work?
A gardener (called a Hybridizer) takes the pollen from one epiphyllum and uses it to pollinate another epiphyllum. Then the waited game starts. First a seed pod develops and the seeds are harvested. Then the plants are grown for five to seven years before the plants produce blooms. Then the hybridizer has the opportunity to see if any of the plants have produced a truly unique bloom. Most Hybridizers wait a few more years to see if the plant is robust and produces a consistent bloom. After that time the hybridizer chooses to register or not and gives each plant a name. Then comes the time to provide cutting to as many friends as possible so that the plant the Hybridizer has developed will live on. If this last step is not done then the plant while registered becomes lost to the world of epiphyllum gardeners.