Afghanistan is a complicated place, with a wide range of cultural and ethnic backgrounds within the confines of a largely imagined state border.
Essentially, in the 1950s through to the 1970s, the country shifts from a kingdom to a communist republic, and seeks closer ties with the Soviet Union. However, several factions and continual infighting mean that power struggles regularly occur, and the Soviets are sometimes unsure of who to support.
After a series of coups and assassinations, Soviet forces cross into Afghanistan on December 24th 1979 to support the new government of Babrak Karmal. Widespread opposition eventually coalesces into what is called The Mujahideen: essentially Afghan rebels fighting against the USSR and central government forces.
Note: Mujahideen and Taliban are not the same! This is common knowledge to those familiar with the conflict and recent history of Afghanistan, but some circles are unable to delineate between the different entities that become Taliban, Mujahideen, and Al-Queda. The Mujahideen were a loose band of many factions fighting the Soviets, the Taliban were an Islamist fundamentalist group that eventually form a government in the 1990s, while Al-Queda begins as an Islamist Anti-Soviet faction before changing to target the US in the 1990s, culminating in the 9/11 attacks.
The Mujihadeen receive funding and weapons from the west, specifically through the CIA. The war drags on for the Soviets with mounting casualties and limited successes.
The Karmal government is effectively ended by the Soviets, and Mohammed Najibullah of the secret police is placed in power instead. Gorbachev comes to power in the USSR and renews focus on Europe, beginning plans for a withdrawal.
After a full decade, the USSR is exhausted by the counter-insurgency and mercurial nature of the Afghan conflict, and withdraw in 1989.
In 1992 the Najibullah government collapses and we begin to see the civil war that will result in the emergence of the Taliban.
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For a very digestible history of Afghanistan I recommend this timeline from PBS
For a more detailed rundown of various points in the Soviet-Afghan war, you can always start with the wikipedia article or check out our reading list