Listening to LiveATC there are references by various ATC entities (towers, approach, etc) to Company. To what does this refer?
Generally when used in ATC movement instructions the word "company" means "The other aircraft operated by your company".
In other cases it can mean "the airline (company) you work for" (such as being instructed to taxi to a ramp and "call company for your gate assignment" - the distinction is made based on the context of the request or instruction).
Note that this is not standard phraseology: despite being widely used the term
It means, "The same airline as you".
The typical communication is like:
It helps the pilot identify the airplane out in front of him. For example, if he spots a Red and Blue Southwest plane, he knows he's not looking at the proper plane! He knows he needs to be looking for another US Air paint scheme.