Sadly, I cannot answer about artillery strikes, although my guess is that this what interests you the most. But i can elaborate about air strikes. For this discution air strikes are attacks on enemy targets in proximity of friendly forces, in battle. Air intrediction, and strategic bombing are not in the scope of this answer.

Air strikes during WWII were directed by the pilots themselves, by visually aquiring a target and bombing it. During the later stages, especially with the american forces, pilots found they have difficulty recognizing enemy positions especially considering that in the most part pilots had no radio communications with ground forces. One solution was puting a pilot with the ground forces that would direct air strikes from the ground, later on pilots would take a light liason aircraft and use smoke rockets to point the strike aircraft to the target they allocated to them, these became the forward air controllers. This system was quite successful and was the main means to directing strikes in the korean, vietnam wars and indeed to this very day.

During the korean war fighter bombers had sights that could compute bullet trajectory mainly, the bomb and rocket trajectory had to be estimated by the pilot. In the vietnam war, more advanced avionics enabled calculation and display the bomb trajectory (what Today is reffered to as CCIP mode), later on even more advanced avionics enabled automatic release of weapons onto a preselected point, what today is known as CCRP mode. The CCIP mode being the main tool for air strikes.

During the vietnam war aerial artillery was used in the form of gunship aircraft like AC-47 Spooky that could fire onto enemy positions with miniguns. and later on the AC-130 Gunship II That included combinations of smaller and larger caliber guns directed by an analogue fire control computer, Infra red camera, and a solid state laser pointer.

even with the introduction of precision guided laser bombs late in the vietnam war, attack aircraft used predominantly gun strafing and unguided rockets and bombs to do the job.

Rather similar to WWII. Using maps with grid coordinates (Military Grid Reference System). The person requesting fire can either give the grid coordinate he wants targeted, or his own grid coordinate, along with the azimuth and approximate distance (Edit: You can also give distance and direction from a known, pre-designated reference point). The Fire Direction Center (FDC) knows the grid coordinate of each gun in the firing battery. Using some trigonometry, they then compute the azimuth and distance from each gun to the target, and pass that information to the gunners, along with the number & type of rounds to be fired, the elevation of the gun tube and number of powder bags, as well as the time to fire.