The Great Game was an intense rivalry between the British and Russian Empires in Central Asia, beginning in the nineteenth century and continuing through 1907. Britain sought influence or control in much of Central Asia, to buffer the "crown jewel" of its empire - British India.
Tsarist Russia, meanwhile, sought to expand its territory and sphere of influence, in order to create one of history's largest land-based empires. The Russians would have been quite happy to wrest control of India away from Britain, as well.
As Britain solidified its hold on India (including what is now Myanmar, Pakistan and Bangladesh), Russia conquered Central Asian khanates and tribes on its southern borders. The front line between the two empires ended up running through Afghanistan, Tibet and Persia.
Britain's attempts to conquer Afghanistan ended in humiliation, but the independent nation held as a buffer between Russia and India. In Tibet, Britain established control for just two years after the Younghusband Expedition of 1903-04, before being displaced by Qin China. The Chinese emperor fell just seven years later, allowing Tibet to rule itself once more.
The Great Game officially ended with the Anglo-Russian Convention of 1907, which divided Persia into a Russian-controlled northern zone, a nominally independent central zone, and a British-controlled southern zone. The Convention also specified a border line between the two empires running from the eastern point of Persia to Afghanistan, and declared Afghanistan an official protectorate of Britain.
Relations between the two European powers continued to be strained until they allied against the Central Powers in World War I.
The term "Great Game" is attributed to British intelligence officer Arthur Conolly, and was popularized by Rudyard Kipling.