First off, let me begin by saying that, as a Christian, I recognize every Christian believes their faith aligns with their political beliefs. That is a given, and isn't likely to ever change. And, I will begin by saying that Christ never actually defined a Christian political program. But there were points in His ministry where he encountered governmental policy.
Secondly, I am writing this because so many, especially Republicans, have lost sight as to what the ideologies actually mean.
I will start with liberal political ideology. Liberals generally believe, as society evolves, the role of government should expand to accommodate this role. The problem with this in American society is that we have a Constitutional Republic whose government should be operating within the narrow constraints of the Constitution. Since constant expansion of government requires operation outside these constraints, Liberals tend to embrace a fluid Constitutional interpretation method called "Loose Constructionism" or "Living Constitution".
According to Wikipedia, Loose Constructionism is... "the claim that the Constitution has a dynamic meaning or that it has the properties of an animate being in the sense that it changes. The idea is associated with views that contemporaneous society should be taken into account when interpreting key constitutional phrases." That is the summary of liberal ideology. Times change, and the role government plays in our lives should change, and expand, with it. This is why liberal ideology is often referred to as "progressive" ideology.
Expansion of government, just like the expansion of anything, requires additional resources. It is for this reason that liberals generally push for higher taxes upon its. In fact, the entire economic system of liberals proposes a complete government regulated economy, eventually trending toward Socialism, the economic model embraced by Communism. The closer a society gets to Socialism, the closer that society will be to putting an end to any free-market Capitalistic enterprise, as well as any private property rights. Everything essentially becomes property of the State, and the citizens are basically wards of the State. This is why Liberalism is called a Collective ideology, as it basically nullifies the whole idea of an individual having sovereign rights to their own personhood.
In America, liberal ideology is most commonly represented by the Democratic Party, and more extreme cases are represented by various third parties, like the Green Party or the Communist Party.
Conservative ideology isn't necessarily the polar opposite of liberal ideology. Conservatives believe in the diminished role of government, preferably to its Constitutional restraints as it was originally written and interpreted it by those who framed it centuries ago. Any expansion of government can be accomplished through Constitutional Amendment, and even that is subject to interpretation by the Supreme Court. The passage of a Constitutional Amendment is difficult, facilitating its precise understanding before passage. It is for this reason that Conservatives are often regarded as "Traditionalists", contrary to the Liberal title of "Progressives".
In general, Conservatives embrace Capitalism, and allowing the natural mechanisms of a free-market, to run unimpeded by government intervention. While Liberals claim such a system creates massive income inequality and potential class abuse, Conservatives claim that Capitalism drives innovation, controls market prices, and improves the overall living conditions within any given society wherein it is allowed to work. Conservatives generally allow some regulation of the free-market, to discourage both monopolies and injustices perpetrated upon the working class.
In America's past, the Republican Party used to represent Conservatism. But over the last four to five decades, this has changed profoundly. Traditionally, Democrats embrace the expansion of government, while Republicans embrace Conservative idea of reducing the size of government. Today, government expands under both Democratic and Republican government control. The only measurable difference being, with Republicans, it grows slightly slower.
Conservatism is more nuanced than Liberalism. With Liberalism, it is far simpler to understand because they will always be for more government control, more government regulation, and more government dependency. With Conservatism, there is always the question, if government is too big, how much fat can we trim before it starts affecting society in a negative way?
Take, for example, Social Security. The Social Security Program was part of Lyndon B. Johnson's, a liberal Democrat, "New Deal" programs, which expanded government exponentially. From a "original intent" Constitutional Interpretation, the whole idea of government taking a portion of your wages (without your express consent) and holding the funds (actually using them to fund other government expansions) until you are retirement age, isn't a enumerated power given to the Federal Government by the Constitution. But today, nearly a century later, the mere mention of ending the Social Security programs will be met promptly with howls of anger. Too many people are now reliant upon the government program to ever allow its demise.
That is the Conservative dilemma. Once an expansive government program has been installed, and a portion of the citizenry has become dependent on it, you cannot repeal it without appearing the "bad guy". It doesn't matter if it passes Constitutional muster from a "original intent" standpoint.
So, how does a Conservative reduce government without getting dirty? This is why Republicans have had such a hard time clearly defining a limited-government platform for the last three or four decades.
Libertarians are a different sort. They are usually either anarchist or minarchist. No government, or extreme minimal government. For Libertarians, the choice is clear. The individual takes precedence above the collective. If government is to exist, it isn't to interfere with individuals at all, but to protect individual rights. That is different than protecting individuals. If government properly protects the rights of an individual, the individual can protect themselves. This is why Libertarians are so fond of the Bill of Rights. While most Libertarians disagree on the potency and efficacy of having a Constitution to constrain government, almost all of them see the Bill of Rights as a list of something that should be protected. The Bill of Rights, the right to life, free speech, to bear arms, to private property, to practice, or abstain from practicing, any religion, etc, is natural, paramount, given to us by our Creator, and inalienable. Government, if it exists at all, should only exist to protect these rights. Often Libertarians cite the oath taken by Federally elected officials. They swear to protect and defend the Constitution. Notice, their oath isn't to protect and defend the people, but the Constitution. The Constitution is an ideal.
Of course, most presidents go beyond their oath within a few days of taking it.
Economically, Libertarians embrace Laissez-faire. According to Wikipedia, Laissez-Faire is "an economic system in which transactions between private parties are free from government interference such as regulations, privileges, tariffs, and subsidies." In other words, a genuinely free-market, where one individual can enter into a contract with another individual without interference.
Libertarians regard taxation as theft, since it is our money taken without our express consent. They also are against the passage of victimless crimes. Libertarians call this the "Non-Agression Axiom". If an activity does not produce a victim, it shouldn't be prohibited. This is why Libertarians are sometimes seen a being proponents of the legalization of recreational drugs and even prostitution. And once our western mindset can get beyond the cognitive-dissonance of legalizing these things, it becomes easy to see why they should be legal. Sex between two consenting adult individuals isn't illegal, why should sex between two consenting adult individuals for money be illegal? Alcohol isn't illegal, why should the use of any recreational drug be illegal?
Libertarians also believe that any government control is a step toward Authoritarianism, more commonly called Statism. The opposite being, of course, Anarchy. Don't let the westernized images of Anarchy you see on television throw you off. Anarchy of the Libertarian type was best summed up by the famous author of "The Hobbit" and "The Lord of the Rings", J. R. R. Tolkien, who said, "My political opinions lean more and more toward Anarchy... the most improper job of any man, even saints (who at any rate were at least unwilling to take it on), is bossing other men. Not one in a million is fit for it, and least of all those who seek the opportunity."