My guess is that you are out of refrigerant.  There is a pressure
sensor that will allow the compressor to run if there is enough
refrigerant in the system.  It will save the compressor from self
destructing if a leak occurs and the system goes empty.
 You could try bypassing the sensor just to see if that is the
problem.  If the compressor turns when the AC is turned on and the
switched is jumped than low refrigerant is your problem.  The pressure
switch is on the low pressure side of the compressor (un-insulated
tubing probably near the tank with the view finder like on my 91
Accord DX).  If you see/hear the compressor turn stop the test so you
don't do any damage to the compressor.
 Finding the leak and filling it is another problem.  Possibly best
done by a professional with the tools and experience.  It's tough to
find a lead you can't see.  There are testing, sealing,  and charging
kits sold for the do it your selfer at automotive stores, Walmart,
K-Mart...  If you have a small leak, AC was previously not as cold as
it should be and getting worse over time,  you might try some sealer
and a charging kit.  It will probably cost about $50 for both.  
 Your system is most likely uses the R-12 not the newer R-134a gas.
Make sure you get a retrofit kit that allows using the R-134a in a
R-12 system.  It comes with special oil and fittings.  I charged my
'91 two times in the last 3-4 years using the sealer and a retrofit
kit.  A lot cheaper than the $300+ a few places wanted to fill it.  Of
course plus parts they would need to replace.  I bet that would have
added a lot more to the cost.  No way a car this old with a lead would
not need parts!

 Good luck!