The plant pictured is Schlumbergera truncata, or a hybrid with this species. It is a type of epiphytic cactus commonly known as Crab Cactus, or Thanksgiving Cactus in the U.S., as it often begins to flower in November, at about the time of that U.S. holiday.
I find that my plants do start flowering in November, but usually reach fullest bloom around Christmas time.
However, somewhat ironically, there is another species, less commonly grown, that is the 'true' Christmas Cactus (Schlumbergera russelliana), even though it is more likely to be in full bloom in January or February. It does not have the 'crab-like' projections on its stem segments like S. truncata.
Both species have a number of cultivars, that flower dramatically in a wide variety of colors and patterns, including white, pink, red, magenta, orange, yellow, and multi-color. There are also hybrids between both species (S. x buckleyi), that may be intermediate in character, though often the traits of one species dominate. Other Schlumbergera species, such as S. opuntioides and S.orssichiana are also sometimes utilized in hybridization efforts.
There is yet another species known as the Easter Cactus (Schlumbergera gaertneri, formerly Hatiora gaertneri or Rhipsalidopsis gaertneri). As its common name suggests, this species blooms in Spring, though usually fullest bloom is later than Easter.
Again, some of the cultivars are hybrids, usually S. x graeseri, which is a hybrid of S. gaertneri and S. rosea.