What ATC entities does a commercial airliner communicate with?
This answer details the procedures used in the US. Other countries will be similar, possibly with different names for some positions.

When preparing for a flight, after their flight plan has been filed, the pilots will first talk to the clearance delivery controller. This controller clears their planned route and altitude. At larger airports this may be done via text rather than voice. The pilots may also call their company dispatch to get the route or fix any problems.

If there are significant delays in effect at the airport, the pilots may have to talk to the metering controller. This controller is responsible for holding aircraft at their gates. Weather at departure or arrival airports may reduce the number of aircraft the airport can handle. There is no sense in sending the aircraft onto the busy taxiways if they will just have to sit there and wait for weather or traffic before they can depart.  The metering controller will eventually hand the pilots off to the next controller when the aircraft is finally cleared for departure.

Again at larger airports, the next contact will be with the ramp controller. This controller is responsible for a ramp area near the terminals. The controller will let the pilots know when it's safe to push back, and which way they should turn at the end of the push back. After starting their engines, the pilots will receive instructions for exiting the ramp.

Once pilots reach the edge of the ramp area, they will call the ground controller. This controller is responsible for managing traffic on most of the taxiways. The pilots will receive instructions for taxiing to their departure runway. If this route takes them across an active runway, the ground controller will have to clear this with the tower controller, or possibly have the pilots switch to the tower controller for permission to cross the runway. At small airports, the ground controller handles all of the previous positions as well.

Upon reaching their departure runway, the pilots will switch to the tower controller. The tower controller is responsible for the active runways. Busy airports may have multiple tower controllers managing different runways. The tower will clear the pilots for takeoff, ensuring proper separation between any other arriving/departing aircraft.

Shortly after the aircraft starts climbing, the tower will hand the pilots off to the departure controller. Depending on the location and traffic levels, this could be the same controller and frequency as the approach controller. This position is part of the TRACON (Terminal Radar Approach Control) facility. High traffic areas will have their own separate TRACON facility, while in lower traffic areas this will be combined with a tower location. In other countries this is referred to as the Terminal Control Center (TCC).

The approach controller's responsibility is routing aircraft leaving airports in the area to avoid conflicts between aircraft, but often the pilots will just fly the published departure procedure. The departure controller receives reports of weather such as precipitation or icing and passes this on to other aircraft.

Upon reaching a certain altitude, the departure will hand the pilots over to the center controller. This position is part of an ARTCC (Air Route Traffic Control Center), often abbreviated to "center". The US has 22 ARTCCs, each further divided into different sectors. Sectors are consolidated between controllers depending on traffic levels. Other countries refer to this as an Area Control Center (ACC).

The center controller is responsible for traffic at higher altitudes, primarily enroute traffic. The aircraft may pass through multiple centers, and each center may have different sectors controlled by different controllers. The pilots will be handed off between these controllers. Pilots must request altitude or course changes (for efficiency or weather reasons), and controllers may issue changes for traffic reasons. Controllers also handle pilot reports of conditions such as turbulence.

If the flight is crossing large areas of ocean, the flight may be handed over to an oceanic controller. These controllers do not always have radar coverage of the aircraft they are controlling. They rely on the aircraft following their published routes and altitudes, and providing regular position reports. The lack of radar means there are also more strict rules about deviations from flight plans.

As the pilots get closer to their destination, they will start on their arrival procedure. This usually specifies altitudes and possibly speeds, so the pilots will need to plan when to start their descent in order to meet these restrictions. The center controller will know the procedure and instruct them to follow it, and possibly issue deviations as traffic requires.

At some point before arriving, the pilots may also contact their company at the arrival airport. They may need to coordinate passenger needs such as wheelchairs or medical/security issues, or communicate maintenance needs. They will also coordinate what gate they are scheduled to use. This can also be done via text, such as through ACARS.

When descending out of center's airspace, pilots will be handed over to the approach controller. This position is part of a TRACON and handles arriving traffic. Depending on the location and traffic levels, this could be the same controller and frequency as the departure controller. The pilots will continue to follow the arrival procedure, again with any deviations issued by the controllers. The departure controller receives reports of weather such as precipitation or icing and passes this on to other aircraft. The controller may also issue information about weather observed on radar, although each aircraft also has its own radar onboard. The approach controller will then provide vectors to the pilots as needed to get them to an approach procedure. The approach controller will then clear the aircraft to fly a certain approach to a certain runway.

Once the aircraft is established on the approach a few miles from the runway, the pilots will be handed over to the tower controller. The tower will clear the aircraft to land. If the aircraft has to go around, the pilots are sent back to approach to be sequenced back onto the approach. The tower controller also receives reports of wind shear and at what altitude pilots break out of the clouds, to pass on to the next aircraft.

After exiting the runway, the pilots will change to the ground controller. As before, the ground controller issues taxi instructions to get the aircraft to the appropriate ramp.

Before entering the ramp, the pilots will call the ramp controller. The ramp controller must see if the aircraft's scheduled gate is available. If not, the aircraft will have to wait on a taxiway somewhere. Once the aircraft's gate is free, the ramp controller allows the aircraft to enter the ramp, and park at the gate.