early 14c., "to guide, aim, or direct," from Old French adrecier "go straight toward; straighten, set right; point, direct" (13c.), from Vulgar Latin *addirectiare "make straight" (source also of Spanish aderezar, Italian addirizzare), from ad "to" (see ad-) + *directiare "make straight," from Latin directus "straight, direct" past participle of dirigere "set straight," from dis- "apart" (see dis-) + regere "to direct, to guide, keep straight" (from PIE root *reg- "move in a straight line"). Compare dress (v.)).
Oldest sense in English is preserved in golf (to address a ball). Meaning "direct for transmission, write as a destination on a message" is from mid-15c. Meaning "to direct spoken words (to someone)" is from late 15c. Late 14c. as "to set in order, repair, correct." The attempt (falsely) re-Latinize the spelling to add- began in France 15c. but failed there (the Modern French verb is adresser), however it stuck in English. Related: Addressed; addressing.