Alright then, from rightmost to leftmost:
The Religious Right: Socially far-right to right-center; economically far-right to center. The Religious Right are crazy. Starting like that seems somewhat biased, I know, but it's true. They're crazy. They are the old white men of the party of old white men. They are ideologically convicted on all societal issues to the point that they are unlikely to ever budge, because these issues are brought down to them from God Himself. Cultural conservatives to the max. Powerfully anti-gay, anti-drug, anti-abortion, pro-Israel, pro-abstinence, pro-school prayer. Usually anti-gun control. Conspiracy theories prevalent, but usually based upon Biblical prophecy rather than more "contemporary" threats. Expect all Americans to live under their set standards, and very angry that they lack the singular power to make this happen. Desire a Christian Nation, essentially. Less convicted on economic issues, but their company means that they usually stay more on the right wing than the center. Prosperity gospel followers in particular connect with economic issues in the same way their fellows do with social issues. Surprisingly ecumenical amongst non-liberal denominations of Christianity, as this was essentially how they were founded. The sick man of the Republican Party demographic-wise, but ideologically convicted enough that they have inflated power. In love with the Tea Party, ambivalent about the Establishment and the more right-wing Moderates (like with their congregations, they see these two as on the right path but still not True and Pure as they are), hate the Libertarians and the more left-wing Moderates.
The Tea Party: Socially far-right to center; economically far-right to right wing. Despite the name, they've been around for much longer. The Tea Party just brought them together. Right-wing populism is the order of the day here. A lot of rhetoric is focused on "common sense" over "the liberal elite". Whether they were grassroots or astroturf is not so relevant anymore, because this beast has eaten its fill and gained a life of its own. Somewhat of a mirror of the Religious Right, in that they are more about economics than social issues, but still usually right wing on both. Reaganomics are more holy to them than the Bible. By the way, the name is supposed to be an acronym:
Taxed
Enough
Already. Anti-Public Sector, Anti-Taxation (there are a few exceptions), Anti-Spending, Anti-Foreign Aid, Anti-Gun Control, Pro-Deregulation, Nativist. Anti-Obama in particular, the main source of the Birthers. The more extreme members take it into outright racism, as I'm sure most of you have seen. Conspiracy theorist hotbed, but more secular about it. Prefer "Obama is planning a socialist coup" to "Obama is literally the Anti-Christ". They and the Religious Right are the dark alliance which holds together the Republican Party as it exists today, and keep it firmly in the right as this causes the least strife between them. Like the Libertarians well enough. Ambivalent about the Establishment. Hate, hate, hate the Moderates.
The Establishment: Socially right-wing to center; economically right-wing to center. Poor, poor Establishment. All they wanted was to make the Republican Party powerful again, and look what happened. Neoconservative. It has representation from all groups by nature, but balances out...kind of. As previously mentioned, the Religious Right and Tea Party are the winning elements at the moment. Tries to keep everybody playing nicely and from the party entering a full civil war. Is more cognizant than anybody else how fatal that would be. Focusing on bringing immigration towards the center on the pragmatic (if flawed) grounds that it will give them a boost of Latino and immigrant support. It isn't going well. Anti-Deficit, Anti-Democrat. Tries to keep quiet otherwise, as going further than that is just too damn dangerous. Just changing immigration is making the Tea Party and Religious Right try to kill the whole party! Headache towards other factions, envy towards Democrats, blackout intoxication towards future.
The Libertarians: Socially center-right to far-left; economically far-right to right-wing. They're, well, Libertarians. Most want either a very small or night-watchmen state, except for the anarcho-capitalists, who want no state. They pull in a lot of voters through their social views, even though such candidates are rarely elected. Some will be positively libertine on social views. If you ever meet a Republican you can't make flinch by describing how you want to create a gay harem and marry them to your city, you're probably talking to one of the Libertarians. Anti-regulation, anti-government, pro-marriage privatization, pro-entrepreneurship, anti-gun control, pro-drug legalization. Pro-Ron Paul.
Pro-Ron Paul. Pro-Ron Paul. ALL HAIL RON PAU- Er, sorry. Yes, they gained a lot of their energy from Ron Paul. Whatever you may think of him, he was undoubtedly the leader of the Libertarians. Rand Paul has been...a lot less impressive to them. Occasionally contemplate defecting to the Libertarian Party en mass due to their lack of size and internal allies, but it doesn't do much. Hate Religious Right, like Tea Party,
dislike Establishment, like Moderates.
The Moderates Socially center-right to center-left; economically right-wing to center. Centrists and Liberals, along with all the other assorted minor groups that don't fit in anywhere else. Depressed and yelling "I told you so! You maniacs! You blew it all up!". Except the ones who are really young, and possibly only Republican because their parents were. They are the ones advocating pragmatic, ideologically blind changes to avoid the party's death. Really don't care as much about social issues, and will follow the national viewpoint on such things even if it brings them into the left wing. Most convicted on social issues, but even that isn't much. Pro-drug legalization, pro-gay marriage, anti-deficit, pro-bipartisanship, pro-immigrant, ambivalent-gun control, ambivalent-abortion. Small but growing alongside the Libertarians. Generally called out as RHINOs by everybody but the Libertarians. Unhappy and worried that the party is doomed, giving them a common ground with the Establishment. Hate the Religious Right, Tea Party, and Establishment for laughing as it all goes to hell. Alright with the Libertarians, but short of the kind of alliance the Religious Right and Tea Party have.
So, that's it. These are the factions that Nate Silver suggested for the Republican Party, and I'd link you do it if I could access the New York Times anymore. Essentially, it's like Olympic Rings. Everybody is connected to the Establishment. Religious Right is connected to Tea Party. Tea Party is connected to Religious Right and Libertarians, Libertarians are connected to Tea Party and Moderates, Moderates are connected to Libertarians.