What is the heating process responsible for moving bands of air around Earth?

Well, the specific process would be convection.  The process of convection occurs when hot air, heated by the sun, rises into the atmosphere.  The higher the air goes, the cooler it gets, so the air condenses and becomes heavier.  Since the air is now too heavy to stay in the upper regions of the troposphere, it starts to sink.  This circular pattern is called a convection cell, and is responsible for all the wind production, both local and global, on the earth.  Other processes that play a factor are radiation from the sun, which causes uneven heating of the earths surface, and conduction, which is heat transfer by direct contact.  The earths rotation also causes directional components for winds, which are divided by latitude markings on the earth.  The equator gets the most direct energy from the sun, so it is naturally warmer there.  The poles of the earth get less direct energy from the sun, so it is naturally colder there.