The Q civil war was a conflict between two factions of the Q Continuum that took place in 2373. Although the deeper nuances of the conflict are beyond the understanding of "mere" Humans, the war was apparently fought between a faction interested in the status quo of the Continuum, and a faction interested in disrupting the static order of the Q's existence. The latter faction was led by Q, the being who had previously appeared to the crews of the USS Enterprise-D, outpost Deep Space 9, and the USS Voyager.
The war first broke out following the suicide of Quinn in 2372 (c.f. VOY: "Death Wish") — the first death of a Q in many years, and an event which severely disrupted the sense of order of the Continuum. Although Q had previously been a supporter of the status quo faction, following Quinn's death he switched sides and became a proponent of less order and control over the actions of individual Qs.
One of the side effects of the civil war was a series of supernovae that erupted across the Delta Quadrant within the space of a mere few days; the occurrence of multiple supernovae only a handful of light-years from each other in such a short time span was unheard of, and believed to be caused by the use of some type of Q weapon as part of the civil war.
Seeking to bring an end to the conflict, Q proposed conceiving a child with Kathryn Janeway, whom he had met the previous year. Q hoped that a Q/Human hybrid child would unite the best qualities of both species and restore the balance of order in the Continuum on a level that both sides would accept. However, Janeway refused to accept Q's plan, arguing that, although his plan was a noble one, it could not serve as a "quick fix" for the underlying problems that the conflict was about.
Q and Q mate, as Kathryn Janeway looks on.
Janeway then attempted to negotiate a truce to end the civil war by contacting a leader of the opposition faction, Colonel Q. However, the status quo faction refused to negotiate, and instead planned to end the conflict by executing Q and Janeway. Before the executions could be carried out, however, the pair were rescued by personnel from Voyager, who were assisted by a female Q, an old flame of Q's.
Q then made his proposal of conceiving a child to his old girlfriend, and she accepted. The two touched fingers, and in a small glow of energy created the first new Q in over ten millennia. This newborn child (nicknamed "Junior") was hoped to be a a herald of a whole new era of peace for the Continuum, and brought about a quick end to the civil war. In gratitude for her assistance, Q named Janeway the "godmother" of his new son. (VOY: "The Q and the Grey")