The British Empire was the name used by the United Kingdom during its time as a colonial power.
In the late 1830s until 1842, the Empire went to war with Afghanistan after occupying the country in a gambit to prevent Russian expansion in Central Asia from encroaching on British territory in India. After a disastrous performance from the British Army, a new army invaded Afghanistan to put an end to the war. (PROSE: Mire and Clay) Some years later, the Empire fought a second war against the Afghans. (PROSE: All-Consuming Fire)
Yvonne Hartman of Torchwood One hoped to use alien artefacts to rebuild the British Empire, much to the Tenth Doctor's confusion. This plan was abandoned after the Torchwood Institute's "old regime" was destroyed in the Battle of Canary Wharf in 2007. (TV: Army of Ghosts)
However, the British or English Empire moniker was also used to denominate several alternate variants of the United Kingdom. For the most part, these were usually more extreme versions of the British government, totalitarian states subject to xenophobia.
One of these parallels was visited by the Sixth Doctor and his companion Evelyn Smythe, which came about when a time fissure caused them to appear simultaneously in England in 1903 and 2003. In the 1903 timeline, they successfully thwarted a Dalek invasion of England, but the local authority seized them both and the remnants of the invasion, scavenging the technology and using it to establish the highly xenophobic English Empire, which by 2003 had all but subdued the entire world under its banner. The Doctor and Evelyn were imprisoned in the Tower of London, where Evelyn died of starvation; after repeated escape attempts, the Doctor's guards opted to have his legs amputated. This timeline was destroyed when the Doctor intercepted the Dalek invasion fleet before they had managed to establish a foothold. While he was successful in restoring history to normal, he was unable to fully erase the taint of the Empire, worried some of it still endured in the minds of the English people. He hoped they would dare to look into the darkness and confront it, eliminating the possibility of the Empire's resurgence. (AUDIO: Jubilee)
Another such parallel came to be due to an alternate Sabbath's manipulations, based on The Oracle, a malicious entity from the Time Vortex. It diverged when, on the Oracle's orders, various agents were spread to murder key personnel essential to the development of the computer. This version of the Empire cowed its citizens into submission and enforced its xenophobia by routinely staging acts of terrorism and blaming on minorities, terrified of the idea of change. Its ruling council, the Star Chamber, was convinced by the Oracle to prevent the development of weapons that could threaten its citizens and endanger their decent lives, though the Eighth Doctor saw through the flimsy argument and learned of the Oracle's manipulations. It was destroyed when Alan Turing, who was forced to become the focal point of the entire reality, was killed. (PROSE: The Domino Effect)