late Old English (replacing Old English fers, an early West Germanic borrowing directly from Latin), "line or section of a psalm or canticle," later "line of poetry" (late 14c.), from Anglo-French and Old French vers "line of verse; rhyme, song," from Latin versus "a line, row, line of verse, line of writing," from PIE root *wer- (2) "to turn, bend." The metaphor is of plowing, of "turning" from one line to another (vertere = "to turn") as a plowman does.
The English New Testament first was divided fully into verses in the Geneva version (1550s). Meaning "metrical composition" is recorded from c. 1300; as the non-repeating part of a modern song (between repetitions of the chorus) by 1918.
The Negroes say that in form their old songs usually consist in what they call "Chorus and Verses." The "chorus," a melodic refrain sung by all, opens the song; then follows a verse sung as a solo, in free recitative; the chorus is repeated; then another verse; chorus again;—and so on until the chorus, sung for the last time, ends the song. [Natalie Curtis-Burlin, "Negro Folk-Songs," 1918]