address (v.) Look up address at Dictionary.com
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" (see direct (v.), and 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.
address (n.) Look up address at Dictionary.com
1530s, "dutiful or courteous approach," from address (v.) and from French adresse (13c., from the verb in French). Meaning "power of directing one's actions and conduct" is from 1590s. Meaning "act or manner of speaking to" is from 1670s. Sense of "formal speech to an audience" (Gettysburg Address, etc.) is from 1751. Sense of "superscription of a letter" (guiding it to its destination) is from 1712 and led to the meaning "place of residence" (by 1888). Transferred use in computer programming is from 1948. Middle English had a noun addressing "control, correction" (late 14c.).