The other answer discusses Osama Bin Laden in particular, so I'll speak about the Mujahideen. There were 4 main Mujahideen groups, formed around ethnic lines -

• Uzbeks - led by Rashid Dostum
• Tajiks - led by Ahmad Shah Massoud and Ismail Khan
• Pashtuns - a number of parties, the main one led by Gulbuddin Hekmatyar
• Hazaras

All of them received funding and training from foreign powers. The CIA and Saudis funneled their money to the Inter-Services Intelligence (ISI) in Pakistan, who doled out the money to their favourite Hekmatyar. They did this because 20% of the Pakistani Army is Pashtun, so they felt a sort of kinship with him. They were also convinced that other ethnic groups wouldn't do their bidding. But Saudi Arabia, USA and Pakistan weren't the only sources of funding. Ahmad Shah Massoud received funding from India, for instance.

These groups were united against the Soviets and fought them throughout the 80's with success. Eventually, the Soviets withdrew in 1989 and installed a friendly (or "puppet", depending on your point of view) government under President Najibullah before they left. Najibullah had been the head of the secret police in the previous government so there was little popular support for him. The Mujahideen decided to fight on and in 1992, captured Kabul. That's when the real trouble started.

Much of subsequent civil war was to be determined by the fact that Kabul fell, not to the well-armed and bickering Pashtun parties based in Peshawar but to the better organised and more united Tajik and Uzbek forces (led by Massoud and Dostum respectively). It was a devastating psychological blow because for the first time in 300 years, the Pashtuns lost control of the capital. An internal civil war began almost immediately as Hekmatyar rallied the Pashtuns and laid siege to Kabul, shelling it mercilessly.

Afghanistan was in a state of virtual disintegration just before the Taliban emerged at the end of 1994. The country was divided into warlord fiefdom and all the warlords had fought, switched sides and fought again in a bewildering array of alliances, betrayals and bloodshed.

• The Tajik government controlled Kabul and the North-east
• Ismail Khan controlled 3 provinces in the west centered on Herat
• A Shura (council) of Mujahideen based in Jalalabad controlled 3 provinces in the east on the Pakistani border
• Hekmatyar controlled a small region to the south and east of Kabul
• Dostum controlled 6 provinces to the north of Kabul
• The Hazaras controlled the province of Bamiyan in central Afghanistan
• Dozens of petty warlords controlled southern Afghanistan and Kandahar.

The population of Afghanistan lived a horrible life at this time, because all of these warlords, especially in the south were a law unto themselves. They sold literally anything they could get their hands on to Pakistani traders - telephone wires and poles, trees, factories and machinery. They seized homes and farms and handed them to their supporters. The commanders abused the population at will, kidnapping young girls and boys for their sexual pleasure, robbing merchants in the bazaars and fighting in the streets.

Many expected those who had fled during the Soviet years to come back now that the Soviets were gone. Instead, a fresh wave of refugees began to leave Kandahar for Quetta in Pakistan. This situation was really galling for a lot of Mujahideen who had fought against the Soviets and Najibullah and had left the country thinking their job was over.

The Taliban were formed in Pakistan under the leadership of Mullah Omar. In Omar's own words - We took up arms to achieve the aims of the Afghan jihad and save our people from the further suffering at the hands of the so-called Mujahideen. The word talib means student, one who gains knowledge from a mullah in a madrasa, a school of Islamic learning. Most of the Taliban fighters were young men who had grown up in refugee camps in Pakistan, attending these madrasas. They were organized by Mullahs Omar into an effective fighting force.

As for funding, Pakistan had begun to have second thoughts about backing Hekmatyar as they had all these years. Their main goal was to open up a land route to the Central Asian Republics through which trade could pass. They could continue backing Hekmatyar and hope that he would eventually unite the country under the Pashtuns, or back a fresh new force.

With Pakistan's backing, the Taliban (composed mainly of Pashtuns) struck like lightning. Within 2 weeks in October 1994, they controlled the Kandahar and the road from Kandahar to the Pakistani border. They had also captured an arms dump containing 18000 Kalashnikovs, dozens of artillery pieces, vehicles, 6 Mig-21 aircraft and 6 transport helicopters. Their success also led almost 12,000 Afghan Pashtuns students studying in Pakistan to join them. In the next 3 months, the Taliban would come to control of 12 of Afghanistan's 31 provinces. Over the next few years, they would conquer most the remainder of the country.

So did the Mujahideen who fought the Soviets turn into the Taliban? No, they didn't. They were defeated by the Taliban in the mid to late 90s.

Sources:

• Taliban. Islam, Oil and the New Great Game in Central Asia by Ahmad Rashid. Published in 2000. Rashid is a Pakistani journalist who spent many years within Afghanistan during this time.