I agree, those are buds on your plant so you will soon see the pretty blooms!
These two have the common names of Holiday Cactus, Thanksgiving Cactus and Christmas Cactus:
Christmas Cactus (Schlumbergera truncata) has pointed leaf segments.
Christmas Cactus (Schlumbergera x buckleyi) has rounded leaf segments.
This one has the common names of Easter Cactus and Holiday Cactus:
Easter Cactus (Hatiora gaertneri) has rounded leaf segments.
