Deadmen (デッドマン, Deddoman) are people who are infected with the Nameless Worm which enables them to do extraordinary things with their blood. Most of them were infected during the Great Tokyo Earthquake.
Ganta Igarashi is an exception for he was infected much later by a unique Nameless Worm by the Wretched Egg herself, during the Nagano school massacre. He became a Deadman with a unique ability, that is the key to the Mother Goose System.
During the Great Tokyo Earthquake, pieces of a certain Red Diamond were found. The Deadmen yet to come somehow inhaled or swallowed a piece of the Diamond. The Red Diamond is a capsule for the Nameless Worm and so the Nameless Worm settled in the bodies of the Deadmen and gave them the Branches of Sin. Those Nameless Worms were somehow created or released by the Wretched Egg.