In Justice League: War, we see Cyborg's origin and he immediately joins the Justice League. However, he is quite young. In certain comics, Cyborg is known as a founding member of the Teen Titans. My question is - does Cyborg leave the Justice League and join the Teen Titans? Is he an active member of both teams? How is it that he is equally represented as an prominent member of both teams?
Depends on the specific continuity. In some cases he's a founding member of the Justice League, in some cases he joins later, in some cases he was previously in the Titans, in some he wasn't.
This is no different from other characters: consider the upcoming movie, where the founding members are Superman, Batman, Wonder Woman, the Flash, Aquaman, and Cyborg. In comics the founding members may include any or only a few of the above, with other characters like Green Lantern (Hal Jordan), Black Canary and (especially) Martian Manhunter being named as original members. In the animated version it was Superman, Batman, Wonder Woman, the Flash, Martian Manhunter, Green Lantern (John Stewart) and Hawkgirl.
For the unofficial version in the Arrowverse on TV, you could probably say it was Green Arrow, the Flash, Spartan, Hawkman, Hawkgirl, Speedy, and Black Canary (Laurel Lance). The other unofficial version from Smallville was (using their superhero names) Superman, Green Arrow, the Flash, Aquaman, and Cyborg.
Marvel has the same thing. The original founding members of the Avengers in the comics were Iron Man, Thor, the Hulk, Ant-Man and Wasp, with the Hulk soon leaving and Captain America coming in. Hawkeye and Black Widow join years later. In the movies, of course, it's Cap, Iron Man, Thor, Black Widow, Hawkeye, and the Hulk, while Ant-Man and Wasp are not members at all.
So there is no one single right answer.
As of Justice League: Origin, Nu52 post-Flashpoint, Victor Stone began as a founding member of the Justice League after being maimed by the explosion of the Boom Tube at STAR labs and rebuilt as a cyborg.
Prior to Nu52, Victor was mutilated by a monster that emerged from an experiment in dimensional travel, was rebuilt with cybernetic parts, and joined the Teen Titans. He was offered Justice League membership during One Year Later, but he stayed with Titans East. He did not actually join the Justice League until Cry for Justice.
Just for the sake of completeness, within the animated Superfriends continuity, Victor was a member of the Super Powers Team. Within the Teen Titans and Teen Titans GO continuity, he is exclusively a member of the Teen Titans.