"strike lightly," c.1200, from Old French taper "tap, rap, strike," from a Gallo-Romance or Germanic source ultimately imitative of the sound of rapping. Meaning "to designate for some duty or for membership" is recorded from 1952, from notion of a tap on the shoulder. Related: Tapped; tapping.
"light blow or stroke," late 14c., from tap (v.1). Tap dancer first recorded 1927, from tap (n.) in the sense of "metal plate over the heel of a shoe" (1680s).
"stopper, faucet," Old English tæppa, from Proto-Germanic *tappon (cf. Middle Dutch tappe, Dutch tap, Old High German zapfo, German zapfen). Originally a tapering cylindrical peg (hence taproot). Phrase on tap "ready for use" is recorded from late 15c.
"to supply with a tap," Old English tæppian, from source of tap (n.1). Meaning "to draw liquor with a tap" is from mid-15c. Extended sense of "make use of" is first recorded 1570s. Meaning "to listen in secretly" (1869), originally with reference to telegraph wires. Tapped out "broke" is 1940s slang, perhaps from the notion of having tapped all one's acquaintances for loans already (cf. British slang on the tap "begging, making requests for loans," 1932).