I believe it used to be specified as the weight of air required to combust one pound of fuel. The figure will depend on the fuel grade, if I remember correctly it is usually about 14.5 pounds of air per pound of fuel.
The value of 14.5 is actually a pure ratio, but the “pound” is often still referenced.
This is quite unrelated to the atmospheric pressure at sea level measured in psi.
Fuel consumption was often specified in pounds per horsepower hour. Now the more common units are kg / kW.hr
A diesel engine usually takes in more air than is needed to burn the injected fuel. Only the air needed for stoichiometric combustion is counted for the specific air consumption, the excess is used to achieve the required compression ignition temperature when idling, and to guarantee complete fuel combustion.