According to Greek mythology their mother was Leda. Pollux then was immortal, like his father. Castor on the other hand was mortal. Though some authors say Zeus was his father too, others site Tyndareus, who had relations with Leda the same night. And rather than twins, Leda bore quadruplets.
Greek mythology tends to read like modern soap opera, only with even more sex and violence.
Castor was a horseman, and could tame the wildest horse. Pollux was a boxer. In their last quest they shipped out with Jason and the rest of the Argonauts in search of the Golden Fleece. Castor was killed in battle with Idas, who was in turn slain by Zeus himself with a lightning bolt. Pollux was inconsolable by the death of Castor and petitioned his father to let him die too.