There are a few ways. Timed and proximity were the most common. Timed worked like setting an alarm clock and proximity could also be set to actively seek a target after leaving the muzzle or after some delay.
Time fuzes work by adjusting the lenght of the powder train in them. When fired, they activate and begin to burn along a labrynth path whose length is set by a fuze setting device on the AA gun. This is common for shells of about 3" (75mm) or larger. The way these fuzes were set was each gun in a firing battery received data on the range and altitude of the target from a central fire control station. There was either a box on the gun that you put the nose of the shell into to set the fuze just before loading it (German 88 for example) or, you loaded the shell on a loading tray and the fuze setter was attached to that (US 90mm for example) and the shell was set automatically before it was rammed into firing position. The second means is percussion or direct hit fuzing. This is common to shells smaller than the above where the rate of fire is too high to allow for setting each shell's fuze. The third method (and best) is a proximity fuze. This activates on firing and will detect the target as it approaches it. At the approrpiate distance the fuze activates and detonates the shell for maximum effect. In WW 2 this was used by the Allies in weapons larger than 3" (75mm) like the US 90mm or British 3.7" AA guns or in naval guns like the US 5"/38.
You should add that the Allied proximity fuse was actually a miniature radar set in the nose of the shell which detected the target aircraft as it approached and, when at the optimum distance from the plane, detonated the shell. The resulting splinters usually destroyed the aircraft and/or killed the crew. It only worked when the AA gunners were able to lay their gun so that the shell passed close enough to the target to detect it and pass close enough to achieve optimum range; radar directed FC was usually the best way to achieve this. Proximity fuses were also used in artillery when air bursts were desirable; the radar set would detect the ground surface and detonate the shell when it was a certain height above the surface, spraying the resulting splinters over a wide area. This was often used where troops were dug in but had little or no overhead cover.
Also not to split hairs, but I didn't see the word "guided" used here by anyone. The word "detected" was used a couple of times, and I believe that equates with "sensing" in this context.
Yes, that is the definition of the term "radar". ra·dar (rā'där) pronunciation n. 1. A method of detecting distant objects and determining their position, velocity, or other characteristics by analysis of very high frequency radio waves reflected from their surfaces. 2. The equipment used in such detection. [ra(dio) d(etecting) a(nd) r(anging).]
A proximity fuse did not constitute guidance; it simply determined when the shell was detonated as was the case with any other type of fuse. Guidance involves the concept of mid-course corrections based upon inputs of information relating to the location and other attributes of the target from either on-board or remote sensors.
"Radar set" implies that it has radar guidance. I quoted the term that's applied to WW2 proximity fuzes.
