Why is the Emergency brake light on in a corolla?

If the handbrake light stays on when handbrake is off the easiest thing to check first is the brake fluid level. The light was on for a month on my '96 Corolla before I noticed the low brake fluid level. After topping the fluid up the light no longer came on while driving. Note that you should only use freshly opened brake fluid - stuff that has been open for a while will have absorbed moisture from the air reducing its effectiveness.

If the brake fluid level is OK then the handbrake cable probably needs adjusting (probably loosening) so the light does not come on too soon, or stay on when it shouldn't. Another possibility is the parking brake engagement sensor, a mechanical switch that is engaged when the parking brake lever is activated. It may have gotten stuck in the on position and need to be cleaned or replaced.