Does the Milky Way orbit around anything?

"Orbit" means that an object moves around a point in space on a certain path. Generally, this center point is an object - except in a binary system, it is a point in space called a 'barycenter' around which both bodies orbit. read more

Generalizing the term "orbit" to mean some larger object / collection of objects to which the object in question is gravitationally bound, I'd say that the Milky Way "orbits" the Local Group, which in turn "orbits" the Virgo Supercluster. Beyond that, the expansion of the universe starts to dominate over gravitation. read more

The masses of your two objects - in this case, the Milky Way and Andromeda - determine the point around which both objects will orbit. This is called the centre of mass, and is defined as the point in space that has an equal distribution of mass around it. read more