What is the name for tear drop shaped regions around a pair of binary stars that indicate which areas are controlled by the gravity of each star?

These are the Roche lobes. They define the gravitational areas that are controlled by each, and can lead to three different types of systems. Detached systems have binaries that each have their own lobes: semi-detached systems have one star that has filled up its lobe, and contact systems are binaries that share a common atmosphere.