mid-13c., avauncen (transitive), "improve (something), further the development of," from Old French avancir, avancier "move forward, go forward, set forward" (12c., Modern French avancer), from Vulgar Latin *abanteare (source of Italian avanzare, Spanish avanzar), from Late Latin abante "from before," composed of ab "from" (see ab-) + ante "before, in front of, against" (from PIE root *ant- "front, forehead"). Compare avant.
The unetymological -d- was inserted 16c. on mistaken notion that initial syllable was from Latin ad-. From c. 1300 as "to promote, raise to a higher rank." Intransitive sense "move forward, move further in front" is mid-14c.; transitive sense "bring forward in place, move (something) forward" is from c. 1500. Meaning "to give (money, etc.) before it is legally due" is first attested 1670s. Related: Advanced; advancing. The adjective (in advance warning, etc.) is recorded from 1843.
c. 1300, "boasting, ostentation" (now archaic), from advance (v.). Early 15c. as "advancement in rank, wealth, etc.;" physical sense of "state of being in front" is from 1660s; that of "a move forward or toward the front" is from 1670s. Commercial sense of "something given beforehand" is from 1680s (earlier in this sense was advancement, 1640s). Meaning "military signal to advance" is by 1849. Also "an act of approach" (1670s), hence advances "amorous overtures" (1706).