A GENIE, short for Genetically Engineered Neural Imagination Engine, was a small, man-made creature with the power to grant its owner's wishes.
The prototype was first created in 2375 by Salvatorio Moretti of the Bureau Tygon. A GENIE was a small scaly creature that looked much like a dragon, but had the bill of a duck-billed platypus. They had advanced AI and could easily figure out how to tap into any available power source to grant bigger and bigger wishes, to the point where they almost had no limit.
In an alternate timeline seen by the Tenth Doctor, the GENIE was specifically created to easily provide everything the 24th century society pleased. However, what was intended as a simple and innocent invention quickly got out of hand. Their inventors failed to realise how powerful they truly were. Because of this, and the GENIE's inability to take into account human nature, they quickly began to destroy Earth.
GENIEs became self-replicating in a way, a person merely had to wish for their own in the presence of GENIE, and the original GENIE made one for them. With everyone having a GENIE, power sources on Earth were quickly drained. People also began wishing harmful things on one another, and wishes could not be regulated.
Some GENIEs brought their owners back to Arabian times and became the inspiration for the genie legends from that time period.
Attempts to stop the use of GENIEs proved futile. Laws could not be passed and enforced, as someone would merely wish the law away. GENIEs themselves could not be wished away despite being able to be wished into existence. People considered wishing GENIEs to never have been invented, but, fearing a destructive paradox, abandoned the idea.
An idea did come about though, as a group of people found the earliest make of GENIE they could, made in May of 2375, and used it to find a way around the paradox. Realising the amount of power it would take, they drew energy from the sun and wished that time be turned back to the day it was first created. Thanks to the enormous amount of power at its disposal, it granted that wish, therefore cancelling out the existence of all other GENIEs. It was known, however, that no GENIE could cope with using the power of the sun, and it burnt up. The heat from its destruction set fire to the lab it was created in and destroyed all research into the project. The GENIEs that had been sent back to Arabian times were erased as well, but echoes and ripples of the timeline they came from remained and mystics and wisemen continued to remember them, leading to the present-day genie legends.
However, Vanessa's GENIE still existed since it had transported her to Rome on April 17, 2375. Being the prototype, it existed before the turning back of time and the nullifying of all other GENIEs. While speaking on her phone with a friend about a documentary she saw on television, she wished she could live in Ancient Roman times. Unbeknownst to her, the prototype GENIE was in the room and, with a thunderous roar, both her and the GENIE were transported back in time.
Eventually, during the Doctor and Rose's search for Ursus, they come across the GENIE in the form of the Roman goddess Minerva. The Doctor, sensing its guise, wished for it to show its true form, to which the GENIE had to comply. After dealing with the ensuing chaos, the Doctor and Rose returned Vanessa to her own time and took the GENIE with them until they could figure out what to do with it. It was Rose, however, that found the solution: she wished for the GENIE's freedom.
With a final wish from Rose for the GENIE's sake, the GENIE disappeared from the console room to an unknown place in space and time to find a peaceful existence for itself, choosing whether or not it would fulfil a wish. The first creature it came across was a guinea pig-like alien. Upon asking if it had a wish, it squeaked, and the GENIE produced its wish: a large carrot-like vegetable. The GENIE concluded that it would be happy there.
In the future, Vanessa's father's laboratory was still destroyed, but she didn't remember the Doctor's warnings about not getting her father to try again and contemplated asking him to make her another GENIE as a result instead. (PROSE: The Stone Rose)