U.S.-led invasion and displacement of the Taliban

Prelude to invasion

Taliban press conference in Pakistan after the September 11th attacks, declaring they will not extradite Osama bin Laden without evidence
After the September 11 attacks and the PENTTBOM investigation, the USA delivered this ultimatum to the Taliban:
Deliver to the US all of the leaders of Al Qaeda;
Release all imprisoned foreign nationals;
Close immediately every terrorist training camp;
Hand over every terrorist and their supporters to appropriate authorities;
Give the United States full access to terrorist training camps for inspection.
On September 21, 2001, the Taliban responded that if the United States could bring evidence that bin Laden was guilty they would hand him over, stating there was no evidence in their possession linking him to the September 11 attacks.
On September 22, 2001, the United Arab Emirates and later Saudi Arabia withdrew their recognition of the Taliban as the legal government of Afghanistan, leaving neighboring Pakistan as the only remaining country with diplomatic ties. On October 4, 2001, it is believed that the Taliban covertly offered to turn bin Laden over to Pakistan for trial in an international tribunal that operated according to Islamic Sharia law. Pakistan, recently recast as an ally of the west, is believed to have rejected the offer (even though they still recognized the Taliban).
On October 7, 2001, before the onset of military operations, the Taliban ambassador to Pakistan offered to "detain bin Laden and try him under Islamic law" if the United States made a formal request and presented the Taliban with evidence. This counter offer was immediately rejected by the U.S. as insufficient.
Bin Laden for his part, maintained America's attack on the Taliban after 9/11 was motivated only by its hatred for Islam.

American attack
Shortly afterward, on October 7, 2001, the United States, aided by the United Kingdom, Canada, and supported by a coalition of other countries including several from the NATO alliance, initiated military actions in Afghanistan, code named Operation Enduring Freedom, and bombed Taliban and Al Qaeda related camps.[The stated intent of military operations was to remove the Taliban from power because of the Taliban's refusal to hand over Osama bin Laden for his involvement in the September 11 attacks, and disrupt the use of Afghanistan as a terrorist base of operations. On October 14 the Taliban offered to discuss handing over Osama bin Laden to a neutral country if the US halted bombing, but only if the Taliban were given evidence of Bin Ladens involvement in 9/11. The U.S. rejected this offer as an insufficient public relations ploy and continued military operations.
The ground war was mainly fought by the Northern Alliance, the remaining elements of the anti-Taliban forces which the Taliban had routed over the previous years but had never been able to entirely destroy. Mazari Sharif fell to U.S.-Northern Alliance forces on November 9, leading to a cascade of provinces falling with minimal resistance, and many local forces switching loyalties from the Taliban to the Northern Alliance. On the night of November 12, the Taliban retreated south in an orderly fashion from Kabul. This retreat was so orderly, that on November 15, they released eight Western aid workers after three months in captivity (see Attacks on humanitarian workers). By November 13 the Taliban had withdrawn from both Kabul and Jalalabad. Finally, in early December, the Taliban gave up their last city stronghold of Kandahar and retired to the hilly wilderness along the Afghanistan - Pakistan border, where they remain today as a guerrilla warfare operation, drawing new recruits and developing plans for a restoration of power.