Is the fan on the outside unit blowing out hot air?
If the blower isn't running but the fan outside is, the problem could be the blower motor, it's run capacitor, or the furnace/air handler control board.
If the fan motor is not running, then shut the power off to the unit. Then check all wire connections for signs of damage such as burning and repair as necessary. You should also check the wire connections in the compressor terminal box and repair if necessary.
You can use our capacitor testing guide to check the unit's capacitors and replace as necessary.
Continue your air conditioner troubleshooting by turning the power back on to the unit and observe whether the contactor closes or not. If the unit does not start, look for a reset button. Some units have a high pressure cutout switch that can be reset. Then, use a voltmeter to check to see if you have power to the unit and 24 vac to the contactor's coil. If you do not have power to the unit, you should check the wiring between the electric panel and the unit. If you do not have 24 volts to the contactor's coil, check the thermostat and the low voltage wires from the furnace/air handler to the condensing unit. If you have power to the unit and the coil on the contactor but the unit does not start, the unit's contactor may be bad.
If the compressor starts but the fan does not, the condenser fan motor could be the problem.
If the condenser fan starts but the compressor does not, the unit may be low on refrigerant, the compressor might be too hot, or the compressor might be bad.