Open the control box and see if you have around 220-240V at the bottom of the contactor. The contactor is the switch looking thing in the bottom. If the unit is calling for air it should have a button in the middle that is pulled in. If you are not getting the volts at the contactor then you need to find the electrical problem.
If you have the correct voltage then look and see if the contactor is connected. The middle button should be pulled in. If you have one where you are unable to see the button, or it is not pulled in, Check the low voltage wiring. This is done by checking to see if there are 24V-28V on the side terminals of the contatcor. These terminals are located on the sides of the contactor. (some contactors have both sides of the 24V on one side.) Either way you should read 24-28V when checking across the two terminals. If you do not then you have a problem with the control wiring. (cut with weed eater, chewed by mouse etc.) I know the 24V coil is good because the inside unit is running. If the contactor is pulled in and the outdoor unit still does not run check and see if there is power at the other end of the contactor. (The end the switch breaks off)
If the switch is pulled in (or you read 24- 28V on the side of the contactor, and there is 240 V coming in on the line. Chances are you need to replace the contactor. Disconnect the power and pull one wire at a time of of the old contactor and install it on the new. After all of the wires are on the new one, remove the old contactor and throw it away. Screw the new contactor with all of the wires back onto the unit. restore power. Often times in our part of the country, there will be ants stuck in the points of the contactor. Just cleaning the contactor will buy more time.
Hope this helps.