late 14c., "very brief portion of time, instant," in moment of time, from Old French moment (12c.) "moment, minute; importance, weight, value" and directly from Latin momentum "movement, motion; moving power; alteration, change;" also "short time, instant" (also source of Spanish, Italian momento), contraction of *movimentum, from movere "to move" (from PIE root *meue- "to push away").
Some (but not OED) explain the sense evolution of the Latin word by notion of a particle so small it would just "move" the pointer of a scale, which led to the transferred sense of "minute time division."
In careful use, a moment has duration, an instant does not. The sense of "notable importance, 'weight,' value, consequence" is attested in English from 1520s. Meaning "opportunity" (as in seize the moment) is from 1781.
In for the moment "temporarily, so far as the near future is concerned" (1883) it means "the present time." Phrase never a dull moment is attested by 1885 (Jerome K. Jerome, "On the Stage - and Off"). Phrase moment of truth first recorded 1932 in Hemingway's "Death in the Afternoon," from Spanish el momento de la verdad, the final sword-thrust in a bull-fight.
"lasting but a moment, of short duration," mid-15c., momentare, from Late Latin momentarius "of brief duration," from Latin momentum "a short time, an instant" (see moment).
1690s in the scientific use in mechanics, "product of the mass and velocity of a body; quantity of motion of a moving body," from Latin momentum "movement, moving power" (see moment). Figurative use, "force gained by movement, an impulse, impelling force," dates from 1782.
momentarily, momently. The first means for a moment (he was momentarily abashed), the second from moment to moment or every moment (am momently expecting a wire from him). The differentiation is well worth more faithful observance than it gets; & the substitution of either, which sometimes occurs, for instantly or immediately or at once is foolish novelty-hunting. [Fowler, 1926]
1680s, "of or pertaining to the mind as a subject of study;" see psychology + -ical. In early 20c. the sense gradually shifted toward "affecting or pertaining to a person's mental or emotional state." Related: Psychologically. Psychological warfare "use of propaganda, etc., to undermine an enemy's morale or resolve" is recorded from 1940. Psychological moment was in vogue from 1871, from French moment psychologique "moment of immediate expectation of something about to happen."
The original German phrase, misinterpreted by the French & imported together with its false sense into English, meant the psychic factor, the mental effect, the influence exerted by a state of mind, & not a point of time at all, das Moment in German corresponding to our momentum, not our moment. [Fowler]