The Rani

The Rani was a renegade Time Lady. She knew the Doctor and the Master when all three were young, and became an enemy of the former and an unwilling ally of the latter.

Youth and Exile
The Rani was the same age as the Doctor. (TV: Time and the Rani) Rani was exiled from Gallifrey after some of her lab mice, as a result of an experiment, grew to enormous size and ate the Lord President's pet cat. They also bit the President himself, triggering a regeneration. (TV: The Mark of the Rani)

Presence in Earth's history
While the Rani certainly did not share the Doctor's fondness for Earth (she referred to it as a "miserable planet"), it was the focus of several of her research projects. At some point, she had visited Earth in the late Cretaceous and acquired several Tyrannosaur embryos.

When the test subjects on Miasimia Goria, a planet she had enslaved, became violently restless and uncontrollable as a side effect of her experiments on them, the Rani visited Earth at various points in its history to extract chemicals from the brains of select human specimens. Because the chemicals in question enabled the human brain to sleep, and because the absence of these chemicals made her victims as violent and uncontrollable as those from her previous experiments, the Rani deliberately chose periods of social unrest to visit, using the violence to conceal her presence and its consequences. She visited the Trojan War, the Dark Ages, the American War of Independence, and finally the Luddite riots in the village of Killingworth during the early 19th century where she used the local bath house as her base, posing as the old woman in charge of the premises. (TV: The Mark of the Rani)

The Master and, shortly after, the Sixth Doctor, interrupted her work. The Doctor sabotaged the navigational system of the Rani's TARDIS, trapping the Master and the Rani inside as time spillage caused the Tyrannosaur embryos to grow at a dangerous rate. (TV: The Mark of the Rani)

On Tetrapyriarbus and Lakertya
On the planet Tetrapyriarbus, the Rani made the acquaintance of, and decided to employ, the Tetraps, led by Urak. With them, she invaded the peaceful planet Lakertya and put into motion a complex plan. The Rani abducted eleven scientific geniuses from across time and space, including Albert Einstein of Earth. Finally she decided to "collect" the Doctor and attacked his TARDIS, causing the ship to go through turbulence. The Doctor was wounded as a result, triggering the regeneration into his seventh incarnation.

The Rani channelled the intellects of the geniuses into a giant artificial brain which she believed could find the secret to manipulating strange matter, the key to making the planet of Lakertya into a Time Manipulator in order to correct what she considered to be errors in the universal timeline. Her first target was to be Earth, where she would prevent the extinction of the dinosaurs, creatures whose full potential she felt had never been truly realised. She considered the death of the native Lakertyans a small price to pay.

She used the artificial brain to find the answer as to how to create a lightweight substitute for strange matter. When it was devised she sent a missile containing the substance aimed at a strange matter asteroid. However, the Doctor destroyed the brain and redirected the missile. Urak betrayed her, leading the Tertraps against her, and they placed her under house arrest in her TARDIS on Tetrapyriarbus. The Rani was 953 years old when these events took place. (TV: Time and the Rani)

TBA
TBA (TV:Episode Eleven) (TV:Home)

Personality
The Rani was an evil (or, arguably, simply amoral) scientific genius whose villainy came not from the usual variety of lust for power and suchlike, but from a mindset that treated everything (including morality) as secondary to her research. She was known to enslave entire worlds in order to have a ready supply of experimental subjects and a place to carry out her experiments uninterrupted. Her major interest was in altering the biochemistry of other species.

While she did appear evil, she found the Master to be truly evil and therefore stupid. She also said that his plans were so overcomplicated that if he walked in a straight line he would get dizzy. (TV: The Mark of the Rani) She simply did evil things because she felt it was necessary to her work. When the Sixth Doctor tried to convince her not to experiment on humans, she called them carnivores and asked if they ever thought of the lesser species when they sunk their teeth into pork chops. This shows that the Rani may have had a conscience of some kind, also shown when she was willing to destroy her test subjects because they would've killed the Doctor. (TV: The Mark of the Rani)

Behind the Scenes
Kate O'Mara portrayed the Rani in most of her television appearances – during the characters reappearance, Lara Pulver was cast in the role.

In August 2012, Steven Moffat stated that "he had no reason to bring back the Rani", thus supposedly putting an end to the rumours of her return to the television series, but in the ninth series she made her return.