There are a number of things that could cause a Mazda Miata to fail to start, though with modern cars the most common reason is a dead battery. Generally a well maintained car that fails to start has a problem with the electrical system; a car that hasn't received regular maintenance may have more serious mechanical problems which prevent the engine from starting.
How to Diagnose a Non-Starting Mazda MIata
Attempt to start the car. If none of the dashboard lights come on, the engine doesn't turn over (you will be able to hear it chugging), and the the headlights don't come on, then there is a problem with the electrical system, and most likely the battery.2
Inspect the battery. If there is a foam on the battery then this means it is leaking dangerous acid and you may need a professional to help you. However, if the there is no foam then you can thump the battery posts with your shoe or a rubber mallet. This may create a better contact between the battery posts and cables, and allow the car to start. You may also try to jump-start the car. If this doesn't work, replace the battery.3
If however, the Mazda Miata won't start after installing a new battery but the electrical system is functional you may have a flooded engine. A flooded engine is when the spark plugs are wet and cannot generate a spark. To dry the plugs you need to enable the Clear Flood mode. This is done by depressing the accelerator all the way to the floor first and then continually turning the ignition. This causes the motor to pump air instead of fuel through the cylinders and will dry off the spark plugs. After a few minutes take your foot off the accelerator and take the key out of the ignition. Then put the key back in the ignition and turn the engine on. If your car does not start after this you have mechanical problems and will require a mechanic with custom diagnostic equipment to properly diagnose.