What is an Airworthiness Directive?
Straight from the FAA:


  Airworthiness Directives (ADs) are legally enforceable rules issued by the FAA in accordance with 14 CFR part 39 to correct an unsafe condition in a product. 14 CFR part 39 defines a product as an aircraft, aircraft engine, propeller, or appliance.


When an aircraft or other component is found to have flaws or other issues, the FAA will issue an AD specifying a solution to be applied to the item. The AD must be complied with, often by a date or total time on the airframe. The action required could simply be an inspection (often recurring after, say, 100 hours of time in service), while in other cases requires modification, strengthening, or replacement of parts.

One of the more prominent cases requiring ADs to be issued was the V-tail Bonanza, which had issues with the tail's structural integrity under high loads. 

The cost of complying with an AD falls on the owner/operator of the aircraft. Failing to comply with an AD renders the aircraft unairworthy. It also could incur FAA penalties, but more importantly, it could result in structural damage or failure, and in some cases ultimately cause a fatal accident.