It's an epiphyllum, probably an epiphyllum hybrid (which typically contain other, similar genera, including Disocactus, Pseudorhipsalis and Selenicereus). These are commonly referred to as 'orchid cactus', but they are neither a true cactus nor an orchid :-) They are a semi-tropical succulent related to cactus that produces large, very showy flowers. They do produce an edible fruit (NOT a dragon fruit) but these are scarce on indoor plants. They are epiphytes, meaning they take advantage of other, larger plants - like trees - as a support system, like most tropical orchids do, and do not require soil in their native habitat. For home gardeners, they grow perfectly well in a good quality, fast draining potting mix.

Unless you live in a semi-tropical climate or certainly a frost free zone (Z 10 or higher), these are not hardy outdoor. And they are pendant in habit, with those long branching structures cascading over the side of a container to often a significant length. One normally grows them in some sort of hanging basket structure or on a tall plant stand to show off the plant and its long branches to best effect. btw, the flowers form at the tips of these flattened leaves/branches so any pruning should be done cautiously.

There are scads of websites on epis and if you do a bit of searching, they will provide all the growing info you need (rather than my repeating it all here!!). I find they are very easy to grow and will reward you with some gorgeous - if short-lived - flowers. I've got one setting flower buds now after being moved back indoors from its summer holiday location.

Here's just one example of an epiphyllum website but do not stop with just one!! And be sure to check out any that show photo galleries of the blossoms - they are incredible!!

I live in N.Fl, right on the ocean, zone 9-b, and can't have indoor plants, but maybe I can just bring this one in during frost warnings, along with my desert rose? Thanks for the info. I was totally convinced it was a dragon fruit until I noticed that mine is flat and the DF has 3 or 4 sides. THANK YOU!!!!!!!!!!!

They are a true cactus, albeit a tropically-dwelling one.

Check out Epic Gardening Kevin Espirito has a great video on how to start.