Bombermen are robotic life-forms with flat computer screen-like faces that have eyes within a squared or circular helmet. They possess a roundish body, long limbs and usually balled hands and feet.
In the 1985 Bomberman title, only the original Bomberman existed. Bomberman worked in a factory of an underground empire, but got tired of this boring life and escaped after hearing a rumor that robots reaching the surface become human. This initial storyline was tied to another acquired Hudson puzzle property, Lode Runner - Bomberman even transforms into the character from that game in the end, making it a direct prequel. The paths of both the Lode Runner and Bomberman series have since mostly diverged (Battle Lode Runner, Cubic Lode Runner and Bomberman Act:Zero being notable exceptions where these elements are revisited). The 1983 computer game lacked an actual storyline, but depicted the main character as human.
In Bomberman for the TurboGrafx-16, Black Bomberman was created by Dr. Mitsumori as the second Bomberman unit, though he had a programming defect that made him mischievous. Some of the other Bombermen appear both at the end of the game and in multiplayer, and may be creations of Black Bomberman. Thus, White Bomberman is viewed as the "prototype model" of his people. The English manual added that White Bomberman was also the first robot created by Dr. Mitsumori. Later games would establish the setting as Planet Bomber rather than Earth, and actual humans would become a rarity later in the series outside of handful like Dr. Ein and Mujoe.
A potentially alternate origin is present in the games Pocket Bomberman, Bomberman Wars and Bomberman World. The first two games appear to have Medieval themes and take place "a long time ago", and the latter appears to feature Bomberman in its familiar futuristic setting. The English Bomberman World manual also refers to the events of Bomberman Wars as dating from "our ancestors", suggesting that Bombermen occur in nature and are not artificial beings in this incarnation. However, the first game also appears to feature White Bomberman and the last contains the Wars characters in multiplayer, so it's possible that this was merely an atmospheric choice on part of the game designers rather than a self-contained retcon.
Other than the Bombermen on Planet Bomber, there are Bombermen with alien origins - the first introduced are the Dastardly Bombers, who had an interest in capturing the original Bomberman. Similar to them, a few like Max are humans turned to cyborgs. Some of the foreign Bombermen are intelligent robots created by good or evil people. Others just appear to be born from different planets.