What Does State/province Mean?

The word state or province can be used for a particular area of land in a country. A country can be a large one and to administrate the whole area by one governing body can be a very tough task. So usually the area is divided into several smaller areas for the convenience of administration. One of the smaller areas of land can be called a state or a province.
      
Usually this division happens for administrative reasons but the bases on which a country can be divided into states are myriad. It can be divided on the basis of language or sometimes other cultural characteristics also play a vital role. Sometimes even religion plays an important role while deciding on the division. But this kind of division usually happens only in a very large country and in smaller countries it is only administrative reasons that play a role in deciding the state borders.

If I live in Greenville, South Carolina what would I put down when asked Sate/Province?

It means a smaller part of something like a country