What does it mean when the "check" light comes on my 1996 Honda Accord ?
What's the worst thing that can happen?

Not being able to drive your car, since it won't pass inspection. It could also indicate that something is seriously wrong with the ignition system, where the car could suddenly die in the left lane in rush hour traffic going 65 mph. Such situations can be very dangerous. My Honda never gave me any trouble, until the Check Engine Light came on and the car wouldn't start. It turned out I had to replace the computer. That's the nature of electronics.

Take it to AutoZone and they will hook it up to a code reader for free. It may turn out to only be a bad oxygen sensor, which is affecting your gas mileage. Regardless, the computer has something it wants you to know.

Gave me an idea on what I could do next