From Funk's Word Origins,

The Greek word gramma, meaning a "letter," is the foundation of the Greek grammatike techne, the "art of letters." This passed into the Latin language as grammatica, into Old French as grammaire, and so into English as grammar. For several centuries in England, Latin was the language of culture. The educated classes conversed in Latin and their social correspondence was carried on in that language. The word grammar during that period meant nothing but Latin grammer [sic], which was regarded as the most important of all the subjects in the curriculum. Our own grammar schools were so named because one of their chief aims was the teaching of Latin grammar.