Diagnosis and Symptoms
If your thermostat is on, the air handler in the attic is on, but the compressor outside is not running, chances are you have a bad capacitor.
This is a relatively straight forward DIY job that can be done with some basic tools and electrical know how.

Warning
STOP ! If you are slightly unaware of household electricity and have not done anything like this, I urge you to finish reading this article for information and NOT attempt this yourself.
When in doubt, call a professional. Your life is worth A LOT more than a couple of hundred dollars for a professional service call.

What is the Capacitor ?
The capacitor is cylinder shaped part in the condenser that is a little smaller than a soda can. Its job is to store a small amount of power and help kick on the compressor and fan when the system calls it to be turned on.
Capacitors for residential applications will normally have 3 connections on top. The first one for the Fan, another for the Herm and the last for Power.
In this example, the wiring is as follows:
Blue = Fan
Brown = Herm
Yellow = Power (C)
(Warning: Do not go strictly on color as they may vary with every different system)

Compressors
I have a couple of Bryant units here which were installed and running for the past 15 years. I maintain them by cleaning them at least once a year with simple cleaners and rinsing them out. The filters for the air handlers inside are also changed regularly.

Hot Day, Blown Capacitor
On a really hot day, I had one capacitor go. We came home to a 2nd floor which was at 87 degrees with the air handler running. Naturally, I went out side to see what was going on. My initial thought was that the turn on lead was cut or ripped by an animal.
So after inspection, I decided to open up the compressor to see what was going on (again thinking an animal got in there and chewed a wire).
After turning off the breaker and pulling the kill switch on the outside (I always do both), I proceeded to take out a few screws that hold the cover on.

Under the cover
Once the cover was off, I could clearly see that capacitor had a raised bump. Having played with R/C cars in my younger years and having different lessons over the years, I knew what bad capacitors looked like.

Now the numbers on the capacitors are important. So it is best to write them down or take the unit with you to go find a replacement. You can usually find one online or hopefully you have a local shop which has these parts.
Fortunately, I have a Grainger close by and was in and out in 15 minutes with a brand new capacitor for only $15 ! The guy who helped me had sold a number of these all summer.
Kudos to the Pros
Now, if you are a professional HVAC technician, hats off to you because you folks work in some really hectic conditions. This is probably a common call type of call for you which will usually happen on the hottest days of the summer and require you to go up into attics and roofs if you are handling commercial applications.
So if you are a home owner and not comfortable with this work, definitely help keep HVAC people in business. My cousin does commercial HVAC systems and he busts his butt.

Safety and Zaps
Now capacitors are known to hold a charge, so you need to be careful as you may get shocked here. I grew up learning to have a screw driver handy to touch the terminals to help discharge it. In my life, I haven’t had any significant shocks. (If you know a better way, please comment below)
Watch the wiring
Take a picture of the wires leads and where they are plugged into. After you pull out the old one and insert the new one, you will simply hook up the leads to the same terminals.
Once you have the wiring done and the screws back in to secure the capacitor, you can put the cover back on. Make sure you have all your tools and any loose screws cleaned up as well.

Get ready to cool down
By now, I was pouring sweat. It was a mild 100 degrees or so, so I was almost happy this was done. (I still had the concern that this wasn’t the only problem). Plug in the kill switch next to the compressor and go inside and turn the breaker on.
Turn the thermostat back on and set it to cool. The compressor should kick on like normal and the system should begin to cool down your home again.

Final Thoughts
If it is not working as expected, then it is time to call a professional HVAC company and have them take a look at your system.
Checking and replacing the capacitor is worth the $15 shot to me before calling the cavalry. I actually bought a spare to have just in case it goes again.
Regular maintenance and cleaning of your A/C units will help prevent these types of problems, so it is better to be proactive with your systems. It is a good idea to clean your coils while the power is off. I use Web Coil cleaner and rinse the units with a hose.

Let me know what you think in a comment below. Was this helpful ? Have you done this DIY ? Do you feel more comfortable talking to a HVAC pro ?
Again, Safety first here ! Do not attempt this if you do not know what you are doing.
Thanks for Reading,
Scottie