A Calabi-Yau space, or shape is a particular class of 6 dimensional object, named after the mathematicians Eugenio Calabi and Shing-Tung Yau. These mathmatical objects have become important recently as they are believed to play a central role in superstring theory.

Super-string theory describes objects that have many extra dimensions, described to be 'curled up' within our own familiar four dimensional universe. However these dimensions can not curl up any old way, the equations governing the theory set conditions and limitations on their final geometry. Edward Witten, Philip Candelas, Garry Horowitz and Andrew Strominger proved in 1984 that Calabi-Yau spaces satisfy these conditions.

However, thousands of Calabi-Yau shapes meet these requirements, and it's one of the major goals of string theory to calculate which of these describes string theory in our own universe.

Source: The Elegant Universe by Brian Greene