The dark rot is a corruption of the flesh that Senua contracts after she makes contact with the darkness of Hela near the beginning of her quest. After witnessing a vision of her body being consumed by it, Senua's right arm is infested with the rot until the end of the game.
After Senua's vision, the game warns players that the dark rot will continue to grow on Senua's arm each time she is killed or fails. If the rot manages to spread all the way up to her head, then the quest ends and all progress is lost.
Even without being killed, the rot spreads up Senua's arm on it's own at pivotal moments in the story, covering her entire right side by the time of the final battles.
There has been some debate as to what the warning of the rot's effect means. Testing of repeated death by players in combat and environmentally have failed to trigger the permadeath state that the warning seems to allude to, even after 50 deaths. Story wise, the rot manages to reach Senua's head after her final confrontation with Hela, fulfilling the warning's prophecy that her quest would be over once it did.
The rot's purpose could be interpreted as a physical symbol of the depths of Senua's psychosis, slowly consuming her from the inside until she learns to accept her condition as part of her and not as a curse.