Trade and economic/political rivalry fueled the Great Game.
The Russian Empire & previous polities wanted access to a warm water port that would not be closed by ice in winter, or easily closed by the forces of the enemy in times of war. So even before the Mongol Empire, in pagan times, the Rus nations hankered to conquer Constantinople. In Ottoman times, the Russians also sought the city, and if a few events were different, would have taken it. This also fueled the Russian expansion into Turkestan and intrigues against Iran. The Great Game was present in Central Asia as well as the Turkish Straits.
The British Empire in turn sought to stem Russia's ambitions to keep the balance of power. They sought to keep the Russians out of Afghanistan & India.
"We don't want to fight but by Jingo if we do/ We've got the ships, we've got the men, we've got the money too/ We've fought the Bear before, and while we're Britons true/ The Russians shall not have Constantinople."
With global warming, the geographical impetus of the Great Game will soon be just a relic of the past.
Edit to make the lines about jingoism more readable.
Ultimately the Great Game was about the British maintaining possession of India, the jewel of the British empire. As Russia continued to expand into Asia the British were terrified that the Russians were going to an attempt an invasion to seize the sub-continent. The Great Game consisted of British and Russian military, political, and commercial action, as well as exploration in central asia to respectively protect and encroach upon British India.
While the Great Game itself ended up fizzling out at the start of the 20th century (not too unlike the Cold War), there are two side effects of The Great Game that ended up having massive consequences in the 20th and 21st century.
Rise of the Japanese. Russian encroachment into Asia led them to the doorstep of the oriental powers, China, and Japan. Russia's imperialist goals led them to direct conflict with Japan resulting in the Russo-Japanese War. The victory of Japan thrusted them into the spotlight as a world power. No longer were the Japanese the backwards feudal nation of fifty years prior that could be easily bullied by the west. The victory led the way to Japan's aggressive imperial mentality, leading to their part in ww2.
The Durand Line. With current events, the Sykes Picot Treaty is a hot topic amongst historians, and the Durand Line is its forgotten brother. Establishing the border between modern day Afghanistan and Pakistan, this line split the natural lands of the Pashtuns, the ethnicity that for the most part makes up the Taliban and its affiliates. The Pashtuns of Pakistan were put under the administration of the Indic Pakistani peoples, while the Afghanistan Pashtuns were left to share a government with Tajiks, Uzbeks, Turkmens, and many other ethnicities. This has led to many ethnic conflicts that continue to go on as we speak.
I highly recommend Peter Hopkirk's The Great Game if you are interested in the subject.