The most common reason for something like this is the capacitor on the compressor (the outside part) has gone out. You can do this yourself, you just need to be extra careful if you decide to do so.
Cut off the power to the compressor
You should have a power disconnect switch feeding your compressor. Turn it off. Use a tester to confirm it's off. You'd be amazed how often disconnects have been bypassed after the fact.
Remove the cover to the side where the capacitor is
You should have one corner that is separated from the open area in the middle. Remove the cover and you should see the capacitor inside (almost always a round cylinder with wires attached). Be aware that the capacitor is still charged even with the power off.
Discharge the capacitor
You probably have a C/Common, a F/fan and a Herm(compressor) terminal. Discharge is recommended. I would suggest using a 100 ohm resistor (not recommended to use a screwdriver here). Touch the common to the herm for a few seconds, then do the same for the fan side.
Remove the capacitor
Take a picture of the wires and then remove the capacitor. The connectors push on to tabs. There's typically a strap to hold the capacitor.
Install the new capacitor
If you can't wait for an online delivery, contact a local HVAC company to see if they'll sell you a capacitor directly. Put the wires back on, scrap it back in and turn the power on.
Your compressor should now work. If not, you might need to consult an HVAC company on what else could be wrong.
