How do airline pilots contact their company while in the air?
While enroute the primary communication with company is via ACARS. (Aircraft Communications Addressing and Reporting System)

This is a digital datalink system and is kind of like sending text messages. On modern aircraft like a B777 it operates on HF, VHF, or SATCOM so there is coverage almost anywhere in the world.  

If the aircraft is close to an airport with a company frequency, regular VHF radio communication can also be used.  Some airlines may even have company Peripheral stations (PAL) at certain locations so they can use a VHF radio to talk directly to Dispatch from hundreds of miles away.

In the USA, VHF company frequencies are allocated by Aviation Spectrum Resources, Inc. (ASRI) ASRI Ground Station Administration Service (AGSA)

SATCOM is also be used to make Satellite phone calls so we can talk directly to Dispatch or Maintenance from thousands of miles away. You can use pre-selected company numbers or manually enter any phone number.  

Aircraft without SATCOM can use VHF or HF radios to contact ARINC (Aeronautical Radio, Incorporated) who can create a phone patch to any phone number.  Aeronautical Radio, Incorporated (ARINC)



Source: B777 Aircraft Operating Manual Volume 2 (2.23 Communications)