1935, as visual equivalent of audio, from Latin video "I see," first person singular present indicative of videre "to see" (see vision). As a noun, "that which is displayed on a (television) screen," 1937.
Engineers, however, remember the sad fate of television's first debut and are not willing to allow "video transmission" (as television is now called by moderns) to leave the laboratory until they are sure it will be accepted. [The Michigan Technic, November 1937]
video game is from 1973.
by 1862, in sporting jargon (curling), "to play (a match) again," from re- "again" + play (v.). Of sound recordings (later video, etc.), "reproduce what has been recorded," by 1912. Related: Replayed; replaying.
The noun is from 1895 as "a replayed match" in sports. The meaning "action of replaying" a sound recording, film, later also video, etc., is by 1953.