Air-conditioning unit on the outside won't come on - fan inside works fine. What would cause the air-conditioning unit not to come on. Thank you.

Check the fuses outside and the circuit breaker in the inside electric panel for the outside section..

After checking breakers, I would check Indoor unit float safety switch, sometimes installers
use the float switch to shut all power to thermostat off and others shut down outside unit only.
If indoor unit is in closet or vertical in garage, look at bottom of unit where PVC pipes come out of unit. Look for wire coming off of cap, pry open and check for water. Some have sight glass. If water is in either of these, you can use shop vac to clear drain from outside where
water drains outside. Allso clear any water out of switch itself. If unit is in attic, look for water in emergency pan under unit. Next I would check refrigerant,Freon, Puron. Most newer
and better units are equipped w/low pressure cut out switches, Also look at t-stat wires to make sure there are no disconected wires, next I would check circuitboard. If breakers are not tripped and no water is backed up in drain you should call pro.