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 be wondering about some of the hybrid names. These names are given upon registration of a hybrid. How does hybridization work? Someone took the pollen from one epi and put it onto another epi. Then they waited patiently for a seed pod to develop. Five to ten years after the seeds have been planted, the plants grow and produce flowers. The hybridizer then chooses the best of flowers to register and gives the flower a name.