There is a switch to keep the compressor from running if the pressure is low. This is because the refrigerant carries the lubricant for the compressor; no refrigerant maybe no lubricant, so the safest thing is to shut it down. This switch is a unit with a plug near the sight glass. You can remove the plug, jump across with a paper clip. If it is only low freon (r-134a) then the compressor will come on. <-- Simple trick to see if the system is low or really dead.
Others have mentioned the cooling fan. When the AC is turned on, the two-speed cooling fan should run on high speed. It should do this even if the engine is cold and the fan is not running at all. If the engine is warmed up and the low speed fan is on, when you turn the AC on, it should then run at high speed.
If you can have someone else turn on the AC while your head is under the hood, the compressor clutch makes an unmistakeable "clack" noise when it engages.
I think the fixes are usually simple. (Although I never heard of the circuit board problem!) On my '94 I have only had to change the o-rings on the compressor itself and it runs great.
Warning - the compressor gets extremely hot, not only from being close to the engine, but from compressing the gas. Hot, as in injury to skin if you touch it for more than a few seconds.