Why does a 2001 Corolla run rough after turning on the air conditioner?

The only thing I know about it is, when you turn on the a/c, it makes the engine work harder. In order to work harder, the engine needs more gas. (Just as, to make the car go faster, you have to give it more gas.) So it might be anything involving gas flow: fuel filters, fuel pump, fuel injector, etc. Or it could be a vacuum problem. Or it could be a sensor. Or it might need a tune-up. If you take it to a mechanic, ask plenty of questions until you fully understand what is wrong and why.