Generally it's a combination of the following elements:
1. The Strongman:
Every extremist movement needs a leader, be they a physical leader (Hitler, Stalin, Mussolini, Francisco Franco, Robert Mugabe, David Duke, Saddam Hussein, Mao Zedong, Osama bin Laden, etc.) or a spiritual/idealistic one (Mohammed) whose teachings are widespread and suitable enough to justify The Strongman's control over an activist constituency. The Strongman need not have any particularly outstanding physical or mental qualities, generally they just need high levels of ambition, a suitable political mechanism, ability to control information nationally, and fervent, inspiring rhetoric.
Women can be "Strongmen," it's just most of the successful extremist movements rely on heavily masculine-associated emphases like a warrior culture, aggression towards the stated enemy, and a constant need for war against an external foe. Needless to say, history is in endless supply of men who fit this model, while women of this sort are confined to less successful cultural/political movements like radical feminism.
2. Utopian Vision:
A vision for the future (generally of a nation, with empire implied as manifest) which requires The Strongman to be in power for an indeterminate amount of time (read: Dictator For Life). This can be anything from race purification to worldwide theocratic control to class-free society to an anti-religious secular state. It doesn't really matter what the vision is, it just has to require The Strongman (or his interpreters/followers) to lead it, his advisers to enforce it, and the media to endorse it. "The Greater Good" spawned from the Utopian Vision almost always serves the purpose of the individual strongman, and such is his ability to exert control he need not abide by the rules he sets for others.
Once you have these two things, you have an extremist movement to latch onto. There are basically two kinds of extremists, the leaders who fit into The Strongman model above and the followers who latch on for either a lack of belonging, susceptibility to propaganda, or physical dependence to the state the strongman has created. The "leaders" will try to jockey for position within the movement, with some trying to unseat The Strongman (usually unsuccessfully) and take over their mantle. The followers are just that: they are lead around by The Strongman's rhetoric because they either hold the same general biases toward the Utopian Vision outlined above, are physically dependent on the social state created by The Strongman, or in the absence of either, fear The Strongman based on his actions and provide an echo to save their own hide.
It is possible to have an extremism without an established strongman, like in Anarchist extremism, but generally these movements are ineffective, or the entire point to them is to make the advocate The Strongman. The Utopian Vision element still remains (in the anarchist's vision, a world without laws to hold them back).