Shore Hardness is a measure of the resistance a material has to indentation. There are different Shore Hardness scales for measuring the hardness of different materials (soft rubbers, rigid plastics, and supersoft gels, for example). These scales were invented so that people can discuss these materials and have a common point of reference.
Shore Hardness becomes an important factor when considering which mold rubber you should choose for making a mold of your original. For example, you would not chose a rubber with a durometer of 70A to make a mold of a plaster standing ballerina with thin, protruding arms bent at opposing angles. A 70A rubber (as hard as a car tire) is stiff and would not offer enough flexibility to extract this model without breaking it. A better choice would be a Shore 30A rubber or softer that would offer enough flexibility to easily extract the model.