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An empire was a state which controlled more than one culturally distinct population groups. These were often different planets and/or species. Empires were usually led by an emperor, (TV: Nightmare in Silver) or empress, (TV: Empress of Mars) who could be elected or hereditary.
There were many empires through time and space. Earth itself had several empires in its history. One of it's earliest known empires was the Roman Empire, that centred in the city of Rome. (TV: The Big Bang) Although it eventually fell, (TV: Day of the Moon) it would later come about again as the New Roman Empire, a civilisation that flourished around the year 12005. (TV: The End of the World)
The largest Earth empire was the British Empire, the empire of the United Kingdom which spread across the Earth. (TV: Tooth and Claw, The Empty Child, Empress of Mars, Demons of the Punjab) By 1850, it was considered the most powerful empire in Earth's history up to that point. (AUDIO: 1963: The Assassination Games) The nation of Japan also had it's own empire, which was the prominent power in Asia during the first half of the 20th century. (PROSE: The Shadow of Weng-Chiang) The Empire's ultimate aim was to unite Asia under the Asian Co-Prosperity Sphere. (PROSE: The Face of the Enemy) The Japanese and British Empires fought against each other during the Second World War. While Japan initially succeeded in taking Britain's colonies in Singapore (AUDIO: The Forsaken) and Burma into it's own empire, (PROSE: Just War, The Scales of Injustice) Britain (serving as part of the Allied powers) ultimately proved victorious in the conflict. (PROSE: Magic of the Angels)
Humanity also grew to eventually have galactic empires. By the year 5000000000, The New Earth Empire was formed and governed by humans across the earth. (COMIC: Agent Provocateur) The Second Great and Bountiful Human Empire had influence in the Tri-Galactic in 42nd century. (TV: Planet of the Ood) They later developed the Third Great and Bountiful Human Empire, which described the Eleventh Doctor as "neither great, nor bountiful, nor overwhelmingly human". (COMIC: A Fairytale Life) They also had a Fourth Great and Bountiful Human Empire that was manipulated by the Daleks. (TV: The Long Game, Bad Wolf, The Parting of the Ways) Humanity was also once directly controlled by another alien force, the Master, who formed the New Time Lord Empire, with the Toclafane acting as his army to rule over earth during the Year That Never Was. (TV: Last of the Time Lords)
Earth's empires also did battle with other alien races. One notable example is the Draconian Empire which was the interstellar empire of the Draconians. (TV: Frontier in Space) The Fourth Great and Bountiful Human Empire also fought and lost against the Daleks. (TV: The Parting of the Ways) Emperor Ludens Nimrod Kendrick Cord Longstaff XLI, Defender of Humanity and Imperator of Known Space, led his human empire in battle against the Cyberiad, a Cyberman empire. (TV: Nightmare in Silver) In 1881, soldiers from the British Empire (serving under Queen Victoria, Empress of India) fought against the Ice Queen Iraxxa on Mars, where she attempted to revive her empire of Ice Warriors. The conflict ended with the humans and Ice Warriors forming an alliance. (TV: Empress of Mars)
Species beyond earth also created their own empires.
The Daleks had their own Dalek Empire, a time- and space-spanning empire created by the Daleks. It served and died in the Last Great Time War. (TV: Dalek, The End of Time, The Day of the Doctor) After the war, Davros created a New Dalek Empire, which was eventually slaughtered by the Meta-Crisis Tenth Doctor. (TV: The Stolen Earth/Journey's End)
The Sontarans had their own Sontaran Empire. (TV: The Sontaran Stratagem, The Poison Sky) They held a deep rivalry with the Rutans, (TV: The Sontaran Experiment) who had their Rutan Empire, the hive minded empire of the Rutan Host, ruled by the Rutan Queen.