Lazarus was a time traveler from the distant past.
In that past, a version of Lazarus from an antimatter universe had visited this universe. Unable to bear the thought of a duplicate, Lazarus became insane, and obsessed with destroying this anti-Lazarus.
He pursued anti-Lazarus across space and time, eventually arriving in 2267, where their conflict created vast cosmic effects that drew the attention of Starfleet.
Assigned to investigate, Captain Kirk and the USS Enterprise confronted Lazarus several times, attempting to learn the truth. Lazarus referred to his counterpart as a "beast" and claimed he had destroyed Lazarus' entire civilization. Kirk and Spock soon realized there were two versions of Lazarus.
In a final confrontation, Kirk was accidentally transported to the antimatter universe, where anti-Lazarus revealed the whole story. He confirmed that if Lazarus succeeded in crossing over to the antimatter universe, and there confronted anti-Lazarus, the result would be the destruction of both universes.
Together, Kirk and anti-Lazarus forced Lazarus into the negative magnetic corridor that connected the two universes. Kirk then destroyed Lazarus' spaceship, which also destroyed anti-Lazarus' ship, sealing the corridor, saving the universes – and trapping both versions of Lazarus together for the rest of time. (TOS: "The Alternative Factor")