Simply the stability of a compound in its current state. Normally the stability is measured by free energy, which is a generating function of other thermodynamic properties.
Do you mean thermodynamic stability or thermodynamic equilibrium? Assuming you mean the first one, then stability means that certain derivatives of thermodynamic state variables have to be positive or the substance would change its state. So for a fluid, dP/dV has to be negative (if the volume increases the pressure must decrease) because if this were not the case a fluctuation that slightly increased the volume would lead to a new pressure that further increases the volume and the whole system would expand forever.
Thermodynamic stability of a compound obtains if the total entropy of the compound plus its surroundings would decrease if the compound would decompose into its constituent atoms or at least into smaller molecules. The formation of a compound ALWAYS entails a decrease in configurational entropy, because atoms formerly free to move independently must now move in lockstep as a unified molecule. The compound can be thermodynamically stable only if its formation entails a more-than-compensating increase in thermal entropy, i.e., only if its formation is sufficiently exothermic.