Q2 was a member of the Q Continuum. At some point in the distant past, he misplaced the entire Deltived asteroid belt, an event which has caused him eternal embarrassment. He was later responsible for stripping Q of his powers and banishing him to mortality as a Human in 2366, having grown tired of constantly apologizing to lesser beings across the universe for the various misdeeds and trouble-making committed by Q.
Q2 continued to follow Human-Q's progress aboard the USS Enterprise-D. He later appeared and re-instated Human-Q's powers when the latter Q committed a selfless act to save the Enterprise-D from the attacking Calamarain, deciding that letting him die was not worth the trouble of explaining his actions to the Continuum. (TNG: "Deja Q")
Q2 was played by actor Corbin Bernsen.
In the episode, both he and John de Lancie's character addressed each other as "Q".
Q2 was never referred to as such except in the script; Bernsen also referred to himself as such in the Star Trek 25th Anniversary Special.