"to kick a ball dropped from the hands before it hits the ground," 1845, first in a Rugby list of football rules, of obscure origin; perhaps from dialectal punt "to push, strike," alteration of Midlands dialect bunt "to push, butt with the head," of unknown origin, perhaps echoic (compare bunt).
Student slang meaning "give up, drop a course so as not to fail," 1970s, is because a U.S. football team punts when it cannot advance the ball. Related: Punted; punting.
late 15c., "a battle," a sense now obsolete, an alteration of skirmish (n.). Sometimes also scrummage, scrimish (16c.). By 1780 as "a confused contest, a tussle;" the meaning in rugby and U.S. football dates from 1857, originally "a confused, close struggle around the ball between players." The sense of "a practice session in which offensive and defensive squads square off" is by 1916.