The Taliban fighters are supporters of a group of extreme religious leaders who controlled Afghanistan from the mid-1990s until 2001.
They are followers of Islam but their beliefs are much more extreme than those of most Muslims.
They are fighting because they want their leaders to get back in charge of Afghanistan.
The Taliban want to turn Afghanistan into what they believe would be the world's purest Islamic country.
What was life like under the Taliban?
When the Taliban leaders were in control last time they banned many things, including education for girls, make-up, kite-flying and films.
But some Afghans still took part in these things in secret, risking extreme punishment if they were found out.
The Taliban's strict laws and extreme punishments did make them unpopular with other countries.
But this was not the reason they were attacked in 2001.