An equation of state (abbreviated EoS) relates temperature, pressure, and volume of a substance. Boyle's law, from the 1600s is the original example: that given a constant temperature, the pressure and volume are inversely proportional. Equations of state are examples of constitutive equations (or constitutive relations), equations describing the behavior of a material, The term equation of state is commonly used in some fields, often for gases. The other terms are also commonly used for solids, such as the effects of stress, e.g., when and how it bends.
An equation of state of a gas, in effect, depends upon its partition function.
The terms soft versus hard (or stiff) refer to a characteristic of the equation of state: soft means pressure varies weakly with density and hard or stiff means pressure varies strongly with density.