"upper limb of the human body," Old English earm, from Proto-Germanic *armaz (source also of Old Saxon, Danish, Swedish, Middle Dutch, German arm, Old Norse armr, Old Frisian erm), from PIE root *ar- "to fit together" (source also of Sanskrit irmah "arm," Greek arthron "a joint," Latin armus "shoulder"). Arm of the sea was in Old English. Arm-twister "powerful persuader" is from 1915. Arm-wrestling is from 1899.
They wenten arme in arme yfere Into the gardyn [Chaucer]
"weapon," c. 1300, armes (plural) "weapons of a warrior," from Old French armes (plural), "arms, weapons; war, warfare" (11c.), from Latin arma "weapons" (including armor), literally "tools, implements (of war)," from PIE *ar(ə)mo-, suffixed form of root *ar- "to fit together." The notion seems to be "that which is fitted together."
Meaning "branch of military service" is from 1798, hence "branch of any organization" (by 1952). Meaning "heraldic insignia" (in coat of arms, etc.) is early 14c., from Old French; originally they were borne on shields of fully armed knights or barons. To be up in arms figuratively is from 1704; to bear arms "do military service" is by 1640s.
"to furnish with weapons," c. 1200, from Old French armer "provide weapons to; take up arms," or directly from Latin armare "furnish with arms," from arma "weapons," literally "tools, implements" of war (see arm (n.2)). Intransitive sense "provide oneself with weapons" in English is from c. 1400. Related: Armed; arming.