Plain English Meaning
An AD is an order from the FAA telling owners and operators that something has been found to be unsafe on a particular type of aircraft or part, and exactly what must be done about it. If an AD applies to your aircraft, you must comply with it — it is not optional.
Full Definition
Airworthiness Directives are legally binding regulatory notices issued by the FAA when an unsafe condition exists in an aircraft, engine, propeller, or appliance, and that condition is likely to exist or develop in other items of the same design. ADs prescribe the inspections, repairs, modifications, or operating limitations required to correct the condition. Compliance is mandatory before further flight unless the AD itself permits continued operation under stated conditions.