Ok, we aren't talking about your family asking you to stop by the grocery store pickup some milk on the way home. On most commercial aircraft there is a text based communications system. This is usually the ACARS system, the Aircraft Communications And Reporting System. This display and keyboard is right there usually in the panel, and encouraged to be used in flight.
The ACARS communications start early in the flight. Most airlines participate in the Pre-Departure Clearance (PDC) program where the pilots get the clearance right from the tower on the ACARS screen. The pilot is required to request the clearance before the flight, and the tower will automatically deliver the clearance to the screen.
The ACARS system is also connected to the aircraft maintenance department. The engine and other parts of the aircraft are connected to the computers that are connected to the ACARS system. When the aircraft is in a certain state, an automated message will be delivered to the ground. Engine status will be delivered when the aircraft is in cruise state, and not accelerating (stable cruise report). On ground time will be delivered when the aircraft has main gear down.
Recently, ICAO has standardized the phraseology and the FAA have started delivering ATC messages to the cockpit. The messaging is called Controller Pilot Data Link Communications (CPDLC). There are some limitations to these messages, generally they will be standard communications. Things like "turn left heading 240 degrees", or "climb and maintain 360". These are normal mundane type messages that the controllers say everyday over and over. The ATC screen has templates of these standard messages, where the controller only need to enter the heading and altitude.
The messages must be acknowledged, or they will be assumed to not be received. The pilot can acknowledge the message or say unable. The controller has the option of using the voice to find out more, or offer a better different message.
There is a huge misnomer, Air Traffic Controllers don't actually control aircraft. The controllers offer suggestions to pilots. Pilots can always do what is needed to operate the aircraft safely, regardless of what the controllers are telling them to do. Normally following the controllers directions will be the safest thing to do, but there is always the option.
The ICAO 4444 document Procedures for Air Navigation Services Air Traffic Management has a chapter on the CPDLC messaging.
CHAPTER 14. CONTROLLER-PILOT DATA LINK COMMUNICATIONS (CPDLC).
There is another document 9694 Manual of Air Traffic Services Data Link
Applications that has additional guidance. These CPDLC messages have various levels of urgency, and alerts, and are outlined better in these two manuals.
There is a fear that using CPDLC will prevent pilots from eavesdropping on other pilots. There is a possibility that may occur. CPDLC message are addressed to a specific aircraft, and to a specific ATC center. Normally, when ATC is talking (using voice communications) to the aircraft in an area, they talk on the one frequency that all aircraft can hear. The benefit to that is that if aircraft are near each other, and a command will make another pilot question the intention, the eaves dropping pilot can ask for clarity. Sometimes controllers make mistakes, and pilots can ask. If someone is put on the same altitude and opposite course as another aircraft, the pilot not getting the command make question the controller. With CPDLC addressed messages, other aircraft cannot "hear" those commands.
Mostly the CPDLC systems are in use in the oceanic realm. There is quite a bit of separation going on in that area, and communications has been poor over the ocean. In the past, the oceanic communications, has been over HF voice channel. CPDLC has actually improved the performance of the communication over the ocean.
Long term, some CPDLC messaging will be added to the enroute area. Perhaps to a limited extent, the TRACON will start to get some CPDLC messaging.