There are, apparently, a few different varieties of these fun, easy to grow succulents. I am pretty sure I have two. The one in the picture above is a Thanksgiving Cactus. The most common "holiday cactus." It is a baby from my in-law's plant. It was given to me in full bloom, just busting with flowers from every direction! I have not had such bountiful success, but it may be because I have not "pruned" it.
The other cactus is a later bloomer making me think it might actually be a true Christmas Cactus.
Also, you may notice that the segments (they are not leaves) are more scalloped and less jagged.
|this picture is from after blooming, before pruning|
Here are a few helpful links I found:
the gardener cook
In the mean time I have pruned both plants to encourage more blooms. Pruning these guys is incredibly simple. No tools needed, just fingers. Gently pinch/grab with your thumb and index finger on either side of the joint you want to separate. Pull apart gently, you may need to give a little twist. You will now have this:
Here are the plants after:
Nothing dramatic. Just took a few segments off here and there to promote growth. It should encourage them to grow more dense, and not be so lanky. I may have a dozen or so new plants here in a few weeks! Want one?