Schlumbergera bridgesii
Christmas Cactus or Holiday Cactus
CACTACEAE
This cactus is not to be confused with the Thanksgiving cactus (S. truncata). They are close relatives. and both have many hybrids. These both will flower within weeks of each other.
Both the daylight period and temperature affect flowering. Buds will appear when the night-time temperature goes to 60 degrees F and the daylight is less than 12 hours. Buds will drop if the temperatures get too high. Sometimes there will be more buds if the plant is kept dry from October to early November.
This is a succulent so watering should be on the dry side. Water them and then let them go dry. The soil should have good proportions of sand. Clay pots are ideal, though many are sold in plastic.
For good growth the cactus should have at least two hours of direct sunlight each day. If outdoors in the summer, light shade is best, in other words full sun all day is bad.
You can fertilize once a week in the spring and summer with a fertilizer that is high in phosphorous (the middle number of the rating). Do not fertilize if there are buds present.
Holiday cacti grow to extreme old age. Many people tell stories of having their great aunt's cactus cuttings.
Some questions from Emily's mailbox:
Q: I have a question about my Christmas cactus. I have had the plant for over a year and my mother in law had the plant for a while before me. It always flowered for her, and It hasn't flowered once while I've had it. What am I doing wrong? 
A: The soil must be kept moist during the growing season, but somewhat dry when the plant is resting. Full sunlight and and average room temperatures are recommended. Short days are required to induce blooming. On September 1, the plant should have total darkness from 6:00 pm to 8:00 am until buds form. Or, if the plant is in a cool room (I have read 53 degrees or also I have read below 68 degrees but above 42 degrees) from mid-September to mid-October, flowers will be produced regardless of the day length.