Brake lights stay on even when the key is of the vehicle.

May be a broken switch above the pedal, but could also be the Rear Electric Module (REM), which contains the Stop Lamp Relay. Is your shifter able to come out of Park without stepping on the brake? If yes, first, pull the connector(s) off of the brake light switch to see if this puts them out. If yes, replace the switch. If no, you may need to replace the REM.
Please see my tip at http://www.fixya.com/cars/r6053280-lights_out_lights_not_out_bulbs_ok . These are generic instructions. If you get stuck or need specific instructions for your car, please get back to me with model, year and engine info.
Stop light switch

You have a brake light switch that is located on the upper part of the brake pedal arm. (up behind lower dash) This may be out of adjustment or bad.