Plain English Meaning
The box that tells your glass-cockpit screens which way the airplane is pointing has stopped working. The attitude and heading shown on that display can no longer be trusted, and the system will usually flag it clearly so the pilot knows.
Full Definition
A malfunction or loss of the Attitude and Heading Reference System, the solid-state sensor unit that supplies pitch, roll, and heading information to the electronic flight displays. When the AHRS fails, the affected display can no longer show valid attitude or heading data, and the system typically annunciates the failure with a warning flag, red X, or removal of the attitude and heading indications.