What is an S-mint quarter?

Answer 1

It simply means the quarter was minted in San Francisco.

San Francisco minted quarters for circulation from the 19th century up to 1954. These coins have a small S on the back under the eagle.

In 1968 proof coin production was transferred from Philadelphia to San Francisco so proof coins made since then carry the S mint mark. New proof quarters have the mint mark to the right of the bow in Washington's wig.

San Francisco often made fewer coins than Philadelphia or Denver so coins with that mint mark MAY (but not always) be worth a bit more than those from the other mints. For most common date Washington quarters up to 1954, however, the price of silver is now high enough that those coins are worth more for their silver content than as collectibles unless they're in almost-new or uncirculated condition.