Cars essentially test their own emissions nowadays with a bunch of sensors. If one of these sensors sees something abnormal, it will throw a code/check engine light in the computer. Clearing this code turns off the light, but the computer still sees a fault until it can reset itself by completing whatever driving cycle is required to reset that particular sensor's reading.

The necessary driving cycle depends on the manufacturer and sensor you're trying to reset. For example, one of the parameters for my mustang is to accelerate from 0-45 at least 5 times. You can either do this on purpose or just drive normally and eventually you'll do it anyway.

That's why you'll still fail a state emissions test if you just shut the light off and go. Their reader wont see a clean emissions reading. You have to drive the car for a while after you clear a code to truly reset everything.