What is the name of the area formed by two intersecting circles? When two circles intersect they form an area which is "ellipse-like" in shape.  What is the name of this shape?
The technical term is lens.${}{}{}{}{}{}{}$
It's called a Vesica Piscis, which is Latin for "fish bladder," a reference to the shape of a fish's swim bladders.

Furthermore, for art history buffs, in Italian it is called a Mandorla, which is also the name used for it in art. Mandorlas were used in medieval and renaissance art as a frame for Christ and other heavenly figures, thought of as the shape of the gate to the Kingdom of Heaven.