Eve was a child and has a humanoid body except her skin, eyes and hair colour are red. Her ears were pointed like elves.
Eve's race could read timelines and manipulate time. They had the ability to possess and control humans. Their children liked to play games; Eve controlled the humans she had possessed to play games at the funfair with her so she wasn't lonely.
A juvenile of the species could be strong enough to manipulate objects for a short while but would eventually be burnt out by the energy. Their powers were at least partially linked to their technology, as demonstrated by Eve's Ship absorbing excess energy from her after she tried to control too many people and machines.
Eve's race was being destroyed in a war between two great powers (presumably the Last Great Time War between the Daleks and Time Lords, as she says her planet was "Exterminated"). Their ability to read the timelines had made them a target for elimination; it is not stated which side committed the genocide, although it is presumed to be the Daleks as Eve said they were exterminated. Eve was smuggled onto a ship by her parents, who sent her away to Earth as an evacuee. Her ship crash landed on a beach. Harry Stevens, the caretaker of a nearby funfair, found Eve and, under orders of Ship, kept her in the haunted mine away from civilisation for her to play games until Ship rebooted.
When the funfair was closed down, Eve got lonely and kidnapped and possessed four humans to play with her in the fair. When Rani Chandra came to investigate the funfair on her own, she stumbled across Eve. Ship allowed Eve to possess the humans to keep her amused.
When Rani persuaded Eve to leave the haunted mine, she tried to control too many people and rides, and began to burn herself out through an excess of energy. Helped to the crash site at the beach by Sarah Jane's companions and Harry, her energies were absorbed by Ship and her health was restored.
With Ship's engines repaired, Eve departs Earth to explore the Universe, taking along Harry and Samuel Lloyd. She and Sam later had a son, Adam, who appeared human.