axe n.

"edged instrument for hewing timber and chopping wood," also a battle weapon, 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- "axe" (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] 

The meaning "musical instrument" is 1955, originally jazz slang for the saxophone; rock slang for "guitar" dates to 1967.

To have an axe to grind is from a Sept. 7, 1810, essay in the Luzerne (Pennsylvania) "Gleaner" 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. It was published in a collection in 1815 titled "Essays From the Desk of Poor Robert the Scribe." The story ("Who'll Turn the Grindstone?") has been misattributed since late 19c. to Benjamin Franklin, a mistake continued in Weekley, OED print edition, "Century Dictionary," and many other sources (Bartlett's "Familiar Quotations" has gotten it right since 1870).

1670s, "to shape or cut with an axe," from axe (n.). Figurative meaning "to remove" (a person, from a position), "severely reduce" (expenses) is recorded by 1922. 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 (attested from mid-15c.). Related: Axed; axing.