Mother Box

They got their name from the fact that they act as caretakers to their users, and provide them with love, in a sense becoming a mother to them. They communicate through repetitive pings, which can be understood by their user. Mother Boxes can only be produced by a being of the Fourth World, and not even all of them are able to do it. It is implied that the character of the maker influences the successful creation of a mother box, and that each Box has a fragment of the builder's personality encoded within. Metron stated that each mother box shares a mystical rapport with nature, and that they have a direct link to the Source, from which they draw their power. So, in a way, Mother Box is best described as a computer link to god. Apokoliptian versions also exist, though they are known as Father Boxes and communicate with a "ping". 

One was given to Orion by Mortalla, wife of Darkseid, after his Mother Box was destroyed, in hopes of using the Box's circuitry to turn him towards Darkseid. Upon the destruction of the android known as Gonzo the Mechanical Bastard, Father Time converted its body into an "orphan box" in the shape of glasses, to be wielded against the Mathemagicians of the Anti-Life Equation.

Abilities

Each Mother box can access the energy to perform a vast variety of abilities. To date, mother boxes have shown capable of:

• Summoning Boom Tubes
• Matter transmutation
• Energy manipulation
• Force Field creation
• Healing
• Gravity control
• Levitation
• Emotion Control
• Illusion casting
• Technopathy
• Fire manipulation
• Phasing
• Telekinesis
• Dimensional manipulation
• And many others...

They appear to have a vast, possibly unlimited variety of abilities, but are generally limited in range to the user's personal space.

Notable Mother Boxes

Among the Most famous Mother Boxes is the one in possession of Mister Miracle, who had its circuitry woven into the circuitry of his suit. Orion has a mother Box which is used to keep his emotions, bestial rage and appearance in check; it was lent to Oracle during the Mageddon crisis. The Mother Box of the Forever People, which they use to summon Infinity Man. 

A Mother Box was given to Iron Man in JLA/Avengers by Metron. He used it to predict JLA 's movements, and it interfaced with his suit. A mother Box was used by Captain Marvel to access his lightning when Shazam was on New Genesis. The Motherbox, in possession of Shilo Norman, is the last Mother Box to survive the war and destruction of Apokolips and New Genesis. Another Mother Box was lent to Jack Knight in his quest to reach Throneworld, and the memories and personality of his father, Ted Knight, were encoded as the Mother Box's artificial intelligence. It discussed was progress with Seyg-El, grandfather of Superman, and counseled Jack in his quests to the Legion of Superheroes' time and to Rann.

Grandmother Box

In an alternate future of JLA: Rock of Ages, Granny Goodness at one point invaded the central computer on New Genesis and somehow got fused with it, becoming an insane Grandmother box. It is destroyed in a suicide attack by Wonder Woman. It attacked with firepit blasts and teleportational abilities.
 

Ancestor Box

After striking Batman with the Omega Effect, Darkseid used a Mother Box-esque device called an Ancestor Box. Within this box seems to be some sort of hyperdimensional creature Darkseid refers to as a "Hyper-Adapter". The box seems to be able to warp time and space, accounting for Batman's travels through time during the Return of Bruce Wayne. The Hyper-Adapter may have some connection to the Hyper Fauna fought by Batman in the aforementioned return, or the sentient Omega fought by Shiloh Norman.