Proud owner of a 1997 Cavalier. Cavaliers are known for having problems with the ATS (Anti-theft System). I do have to give Chevy props because they have a system that actually disables the car. Anyway, The ignition switch in a Cavalier has to read the key. The computer has a code that has to either match with the key or it has to complete the code. The best way is to stick the key in the ignition and turn it to the "on" position. DON'T crank it! Just leave it in the "on" position. The ATS light might start flashing. One key thing is that you should listen for the fuel pump to start. Crank the car on. The reason why you should wait after you turn the key to the "on" position is because it will give the computer some time to read the key. If you turn it on and the ATS is activated, then you have to wait ten minutes. Leave the key in the "on" position. After the ten minutes the light should stop flashing. When it stops flashing then you can turn the car on.
Note: When you turn the key to the "on" position, you should be able to hear the fuel pump run. If you can't hear the fuel pump, then the ATS system is ready to shut the car off. If the Anti-theft System is starting to give you problems, then your best bet is to start thinking about a visit to the dealership or a shop. Its a problem that starts off like a bite from a brown recluse spider. It's small at first but then it gets to be a huge problem if not given any treatment.