Online Etymology Dictionary

aww: see aw.
ax (n.): see axe (n.).
axe (n.): Old English æces (Northumbrian acas) "axe, pickaxe, hatchet," later æx, from Proto-Germanic *akusjo (source also of Old Saxon accus, Old Norse ex, Old Frisian axe, German Axt, Gothic aqizi), from PIE *agw(e)si- (source also of Greek axine, Latin ascia).
The spelling ax is better on every ground, of etymology, phonology, and analogy, than axe, which became prevalent during the 19th century; but it is now disused in Britain. [OED]

The spelling ax, though "better on every ground, of etymology, phonology, & analogy" (OED), is so strange to 20th-c. eyes that it suggests pedantry & is unlikely to be restored. [Fowler]
Meaning "musical instrument" is 1955, originally jazz slang for the saxophone; rock slang for "guitar" dates to 1967. The axe in figurative sense of cutting of anything (expenses, workers, etc.), especially as a cost-saving measure, is from 1922, probably from the notion of the headman's literal axe (itself attested from mid-15c.). To have an axe to grind is from an 1815 essay by U.S. editor and politician Charles Miner (1780-1865) in which a man flatters a boy and gets him to do the chore of axe-grinding for him, then leaves without offering thanks or recompense. Misattributed to Benjamin Franklin in Weekley, OED print edition, and many other sources.
axe (v.): 1670s, "to shape or cut with an axe," from axe (n.). Meaning "to remove, severely reduce," usually figurative, recorded by 1922. Related: Axed; axing.
axel (n.): skating jump, 1930, named for Norwegian skater Axel Paulsen (1855-1938). The name is said to be derived from the Old Testament name Absalom.
axial (adj.): "pertaining to an axis," by 1825, from axis + -al (1). Related: Axially.
axillary (adj.): "pertaining to the armpit or shoulder," 1610s, from Latin *axillaris, from axilla "armpit, upper arm, wing" (see axle).
axiom (n.): late 15c., from Middle French axiome, from Latin axioma, from Greek axioma "authority," literally "that which is thought worthy or fit," from axioun "to think worthy," from axios "worthy, worth, of like value, weighing as much," from PIE adjective *ag-ty-o- "weighty," from root *ag- "to drive, draw, move" (see act (n.)).
Axioms in philosophy are not axioms until they are proved upon our pulses. [Keats, letter, May 3, 1818]
axiomatic (adj.): 1797, from Greek axiomatikos, from axioma (genitive axiomatos); see axiom. Form axiomatical is attested from 1580s.
axion (n.): 1978, from axial + scientific suffix -on.
axis (n.): 1540s, "imaginary straight line around which a body (such as the Earth) rotates," from Latin axis "axle, pivot, axis of the earth or sky," from PIE *aks- "axis" (source also of Old English eax, Old High German ahsa "axle;" Greek axon "axis, axle, wagon;" Sanskrit aksah "an axle, axis, beam of a balance;" Lithuanian aszis "axle"). Figurative sense in world history of "alliance between Germany and Italy" (later extended unetymologically to include Japan) is from 1936. Original reference was to a "Rome-Berlin axis" in central Europe. The word later was used in reference to a London-Washington axis (World War II) and a Moscow-Peking axis (early Cold War).
axle (n.): "pole or pin upon which a wheel revolves," Middle English axel-, from some combination of Old English eax and Old Norse öxull "axis," both from Proto-Germanic *akhsulaz (source also of Old English eaxl, Old Saxon ahsla, Old High German ahsala, German Achsel "shoulder"), from PIE *aks- "axis" (see axis). Found only in compound axletree before 14c.
axolotl (n.): 1786, genus of Mexican salamanders, from Spanish, from Nahuatl, literally "servant of water," from atl "water" + xolotl "slippery or wrinkled one, servant, slave" [see Frances Karttunen, "An Analytical Dictionary of Nahuatl"].
axon (n.): "axis of the vertebrate body," 1842, from Greek axon "axis" (see axis).
axonometric (adj.): 1869, from axonometry (1865), from Greek axon (see axis) + metria "measurement" (see -metry).
ay: see aye.
ayah (n.): "native nurse, children's governess," Anglo-Indian, 1782, from Portuguese aia, cognate with Spanish aya, Italian aja, etc., "nurse," from Latin avia "grandmother," fem. of avus "grandfather" (see uncle).
ayatollah (n.): honorific title for an Iranian Shiite religious leader, 1950, from Persian, from Arabic ayatu-llah, literally "miraculous sign of God."
aye (interj.): word of assent, 1570s, of unknown origin, perhaps a variant of I, meaning "I assent;" or an alteration of Middle English yai "yes" (see yea), or from aye (adv.) "always, ever."
aye (adv.): "always, ever," c. 1200, from Old Norse ei "ever" (cognate with Old English a "always, ever"), from PIE *aiw- "vital force, life, long life, eternity" (source also of Greek aion "age, eternity," Latin aevum "space of time;" see eon).
ayurvedic (adj.): "pertaining to traditional Hindu science of medicine," 1917, from Sanskrit ayurveda "science of life," from ayur "life" + veda "knowledge" (see Veda).
azalea (n.): type of flowering shrub, 1753, Modern Latin, coined by Linnaeus from the fem. of Greek azaleos "dry," related to azein "to dry up" (see ash (n.1)). The plant thrives in sandy soil.
Azerbaijan: country name, of unknown origin, perhaps from Old Persian Aturpatakan, from Greek Atropatene, from the Persian satrap Atropates, who ruled there in the time of Alexander the Great; or from local azer "fire" + baydjan (Iranian baykan) "guardian," in reference to fire-worship.
azimuth (n.): "distance of a star from the north or south point of the meridian," late 14c., from Old French azimut, from Arabic as-sumut "the ways," plural of as-samt "the way, direction" (see zenith).
azo-: word-forming element denoting the presence of nitrogen, 1879, from comb. form of azote, the old term for "nitrogen" (from Greek a- "not, without" (see a- (3)) + zoion "a living being;" see zoo); coined in French by Lavoisier & de Morveau because living things cannot survive in the gas.
azoic (adj.): "pertaining to the period of Earth's history before life appeared," 1854, with -ic + Greek azoos, from a- "not, without" (see a- (3)) + zoon "animal," here used in the sense "life" (see zoo).
azotemia (n.): 1900, also azotaemia, from azote "nitrogen" (see azo-) + -emia "blood."
Aztec: 1787, from Spanish Azteca, from Nahuatl aztecatl (plural aztecah), meaning "coming from Aztlan," name of their legendary place of origin, usually said to lie somewhere in what is now southwestern U.S.
azure (n.): "sky-blue color," early 14c., from Old French azur, asur, a color name, from a false separation of Arabic (al)-lazaward "lapis lazuli," as though the -l- were the French article l'. The Arabic name is from Persian lajward, from Lajward, a place in Turkestan, mentioned by Marco Polo, where the stone was collected.