Praktike has a great post on the vagueness around the term "Islamist". Historically, I think Chanad's right in that the term was originally coined by French scholars to denote movements born "out of modernity (not as a reaction to it) … specifically concerned with taking control over the State, rather than perceived social piety." But alas, a term like "Islamist" will always fall prey to ambiguity—that's the nature of one-word terms with broad root words—so while it would be great if everyone could get on the same page and use the term in the same way, that probably won't happen. It's time, I think, to start using compound terms that will help us get our concepts straight. (This is true in general—as soon as a term starts getting fuzzy, it's time to go compound.)

For starters, let's say there are "Radical Islamists". This group includes al-Qaeda, many Wahhabists, Hamas, the now-nearly-defunct Al-Gamaa al-Islamiyya in Egypt, and a whole slew of other terrorist groups and jihadists. These movements have largely revolutionary political aims, seeking to take total control of the State usually by working outside the existing political process—as you'd expect, violence is the preferred option. The new political order will be governed by religious authorities and institute strict sharia. No exceptions. No happy protections for minorities or non-Muslims. No wearing suits and ties. No. No. No. Needless to say, these folks are mostly by-the-letter fundamentalists when it comes to the Qur'an, though some of the interpretations of jihad are rather, er, interesting. See Mark Gould for more on this.

"Political Islamists" or even "Mainstream Islamists" are usually willing to work within the existing political process to achieve its ends. As Chanad says, they're born of nationalism and modernity, rather than strict reactions to both. Hamas is starting to move into this category, as is Hizbullah, and political groups like the Muslim Brotherhood and Jamaat-e-Islami in Pakistan definitely belong here. These folks are also fundamentalists, and would like to see a strict sharia state set up, but a few exceptions can pop up here and there. Democratic institutions can be allowed—you don't have to have direct rule by religious authorities, but rulers should certainly consult those authorities, and they better not deviate too far from Islamic law. There's some question on whether mainstream Islamists would allow some human rights—would, for instance, the Muslim Brotherhood force sharia on Egypt's 4.5 million Coptic Christians? Jamaat-e-Islami, from what briefly I know, seems to take more progressive stances on a few things out of necessity—they need those female votes!—but religion usually bends to political necessity sooner or later.

Let's see. Then you have "Islamic Conservatives," who don't qualify as "Islamists" mainly because they seem to care more about implementing the social/cultural values of Islam rather than building a thoroughly Islamic political order. Many of the Shi'ites in Iraq seem to fall under this category—Ibrahim Jaaferi, maybe even Ayatollah Sistani—although many Shi'ite leaders are mainstream Islamists, and it's not always clear what's what. You can draw a close analogy to the Christian Right here in the United States. And much like the Christian Right, they're not strict fundamentalists, they do a lot of picking and choosing, and it's not the biggest deal in the world if they don't get a strict Islamic state. The government should, however, seek to promote Islamic values at all times and consult with religious authorities, and Islam should be a major guiding force in writing laws. Conservatives are usually willing to compromise on human rights and the rule of law, though mileage varies—whether the Iraqi Shiites will allow, say, a Christian to testify against a Muslim in court is an open question.

Note: If you took a poll of all the Muslims around the Middle East, I'd bet that the vast majority are Islamic Conservatives. In fact, I think the various orders of Sufism practiced in northern Iraq by the Kurds tend to fall under this heading, and Kurdish "secularism" tends to get overrated. (Though it's true that the Kurdish educated and ruling classes appear to be largely secular.) Anyway, the point here is that it's hard to predict how Islamic Conservative voters will vote, because they have a variety of concerns beyond implementing strict sharia. Sometimes they'll vote on "moral values", but sometimes they'll vote on economics, or national security. You can see where I'm going with this....

On the lighter side are the "Islamic Modernists," similar to the "freestyle evangelicals" here in the United States. They're less fundamentalist, tend to believe that the Qur'an implicitly supports women's rights and a variety of other minority protections, and aren't big on imposing sharia or rule by anything other than the good ol' Lockean social contract. I can't think of anyone in Iraq who falls under this category, but you see this around the Middle East in groups like Al-Wasat, which was a breakaway group from the Muslim Brotherhood. These groups are pretty rare though.

By the way, plenty of secular leaders can be devout Muslims in private. You could compare these folks to American politicians like Bill Clinton, but they seem even more remarkable to me since Islam puts a much greater emphasis on the public sphere than does Christianity.

Anyway, that's my broad classification, and it seems to work for me, though with the understanding that lines get blurry. I don't know a whole lot about movements outside of the Middle East, so there could be exceptions here. In particular, some sects of the Deobandi trend, particularly in India, have very puritanical interpretations of Islam but actively avoid politics. Weird. Though obviously other Deobandi groups—like, I dunno, the Taliban?—see things differently.