Before you can understand Mother Boxes, it's useful to take a detour to understand DC's New Gods. About 18 billion years ago, there was a world named GodWorld. As the planet evolved, humanoid beings sprang forth and developed vastly powerful abilities over billions of years. Eventually, they became gods, aka the Old Gods. The era saw these Old Gods achieve immense technological and cultural progress until a civil war broke out. GodWorld fell into internal conflict between factions of gods, culminating in Ragnarok (GodWorld's destruction) and a massive wave of energy (the Godwave) spreading throughout the universe, sparking the development of gods around the multiverse.
The remnants of GodWorld itself eventually turned into two distinct planets. On the one hand, there's the utopian New Genesis, ruled by the virtuous Highfather. On the other, there's the evil and dark Apokolips, where tyranny and war reign. These planets become home to a new set of deities, the New Gods, powerful and immortal beings that rival or surpass earthly gods in their might and capabilities. The New Gods of New Genesis are, of course, generally wise and just. Those on Apokolips, who eventually come to be ruled by the tyrant Darkseid, are devoted to universal conquest and tyranny under his malevolent hand.
Each planet has developed considerably advanced science and technologies in order to achieve their goals, including tech like the Mobius Chair, Highfather's staff, or Orion's Astro-Harness. Of all these incredible technologies, perhaps the most unique and complex inventions of the New Gods are the mysterious Mother Boxes, the sentient supercomputers that can do nearly anything.