Both the Karzais and the Taliban are products of Kandahar, and both are Pashtuns. The Karzais have been traditionally a very powerful and influential family and were fervently opposed to the Taliban, after they emerged from the madrassahs in the refugee camps of Pakistan and seized power.
The Taliban were religious zealots and wanted to impose Islam and Sharia on Afghanistan, which they did when they were in power. They were ruthless in their rule and rigidly applied their version of Islam, no rights or schooling or jobs for women, beards were compulsory for men, no kite-flying or anything else that gave pleasure, alliance with Pakistans ISI (which had recruited and trained them) and the bin Laden led al Qaeda, beatings, imprisonment and beheadings. They were targeted after 9/11 and the war continues.
The killing of Wali Karzai, is a chilling message that anyone and everyone is vulnerable. A vacuum has been created by his death and Pres. Karzai has lost one his most trusted and powerful ally. He has to act very quickly to fill this void but there are limits on what can be done and who to appoint.
Since the Americans have announced the "draw down' and the Canadians and NATO have done likewise and have trumpeted their victory, the Taliban have become more daring and active. This will be heightened and you can expect an escalation. There is also a growing strain between the USA and Pakistan (since the Osama raid) and millions are being withheld from the Pakistani military. This does not help the situation. Pahistan may be less willing to carry the fight to the Taliban, a fight which has been very bloody and the ISI may be up to its old tricks with the Taliban