Surface area is the measurement of the total area (faces or curved surface) of a solid figure.
Surface area is normally talked about in coffee brewing in regard to grinding coffee. The purpose of grinding coffee is to increase the surface area of the coffee bean. When more of the surface area of the coffee beans is in contact with the water, extraction is quicker and more effective, and more of the properties from the coffee bean are absorbed and dissolved into the water.
Surface area might also be discussed in the context of coffee brewing when talking about the surface area of a cup of coffee, for example. If the surface area of the cup of coffee is larger, the coffee will be more diluted and it will be quicker to cool. Hence, smaller beverages, such as espresso, are served in smaller cups.