There are several possible causes for this, and let's go through them in order of cheapest to most expensive:
1) Power failure. Causes: Cooked fuses or tripped breaker in the outdoor unit. Usually this means the thermostat isn't showing a readout (digital thermostat only) but not always. Some systems power the 'stat from the air handler, and of course lots of thermostats have battery backup, and there's always the analog, non-powered thermostat. You didn't tell us what you have.
Solution: reset the breaker or replace fuses. If it immediately trips again, STOP. Call a tech.
2) Failed control transformer. This would also show at the thermostat, again with the caveats listed above.
Solution: test the transformer if possible. Replace if it's failed. Match ratings, don't worry about brands or model numbers. Voltage, AC or DC and volt-amps output are the important criteria that need to be matched.
3) Failed contactor. The contactor is really just a big relay. If it's dead, the outdoor unit won't come on even though the transformer is working and the thermostat is calling for cooling. You can take the control leads loose (TURN OFF POWER FIRST) and test the contactor's activation coil for continuity: if it's got infinite ohms, it's burned out. Or the contactor can be working, but has so much oxidation and damage on its contacts that it cannot pass sufficient power to the motors for them to start.
Solution: replace the contactor. Match ratings. Generic will work fine, just have the same activation coil and amperage rating and you're good.
4) Failed fan motor. If this happens, the compressor will either trip a high head pressure switch, which protects the compressor, or else the compressor will burn out. Hope it's the former.
First test the capacitor that starts the fan motor. These fail more often than motors do, and they're both cheaper and easier to replace. With the power on, the contactor pulled in and trying to make things start, the fan motor may hum. Give it a quick flick with a stick or screwdriver in the proper direction - if it starts, you probably need a new capacitor.
Solution: replace the capacitor. You probably will have a combined cap that runs both the fan and the compressor: match ratings. Fan goes on FAN terminal and the compressor goes on HERM.
If the motor doesn't spin up, and especially if you can test the capacitor and it shows to be working, replace the motor. Note how your motor is mounted because different manufacturers do it different ways and some of them don't swap back and forth well at all.
5) Leaked refrigerant charge. You can't fix this. You have to be a certified refrigerant tech in the US to even purchase refrigerant, let alone install it. (Automotive systems using R-134a are exempted from this requirement, which is why you can buy that stuff everywhere) If the refrigerant leaked out, you should really spend the extra bucks to find where the leak is and fix that, THEN have the system pulled down, evacuated, leak tested, and refilled to factory spec.
6) Dead compressor. Depending on how it died, this may really indicate that you need an entirely new HVAC system, because a burned out compressor can poison the entire refrigerant circuit. But again, this needs a pro.
If none of this made sense or you don't know how to perform the tests described, STOP. It's easy to screw things up worse, or hurt yourself. Call a pro.
Good luck with it.