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 this context it's a convenient shorthand for controllers and pilots: If two aircraft from the same operator are going to the same place on the airport (one may be told "Follow company"), or to assist with visual separation and sequencing on approaches.
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
company does not appear in the pilot/controller glossary or the Air Traffic Controller's Handbook .
It means, "The same airline as you".
The typical communication is:
"Cactus 123: You are cleared to land, 31L. Following Company Traffic on a 2 mile final"
"You are following another US Airways flight, that is currently 2 miles from the runway"
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.