The specific volume of a gas is the volume of the gas divided by the mass of the gas contained in the volume. Specific gravity is the weight of an object divided by the weight of the same volume of water.
Specific heat is the calories per weight of substance.
First, it simplifies our mathematical analysis of some problems by removing the dependence on mass. For example, the equations for the thermodynamic work of a gas contains the mass flow rate as a multiplier. We could carry it along throughout the analysis, but since the mass flow rate is a constant, the math is easier if we just divide it out at the beginning and put it back in at the end. Second, specific properties are an indication of efficiency. For example, two different engines will have different values of specific thrust.