There are many issues where religion and politics overlap. e.g. Abortion, Homosexuality, Environment, Evolution, Celebration of religious holidays in public places, The 10 Commandments in public buildings, prayer in school, ....

There was a lot of talk about the "Religious Right" having a big effect on the 2004 presidential election.
There is certainly a big overlap with conservative Christians and the Republican Party, but the question of where the boundaries are between religion and politics would take up volumes.

These issues also cause controversy within churches. In 2005-2006 there were heated discussions over homosexuals and women in the clergy in the Episcopal and Southern Baptist churches as well as others.

A New York Times Article on July 30, 2006 titled "Disowning Conservative Politics, Evangelical Pastor in Minnesota Rattles His Flock", discusses Rev. Gregory A. Boyd who preached six sermons called "The Cross and the Sword" in which he said the church should steer clear of politics, give up moralizing on sexual issues, stop claiming the United States as a "Christian nation" and stop glorifying American military campaigns. This was after multiple attempts to get him to preach on these topics. Some Topics I've started to document:
Creation vs Evolution
Religion and the Environment

War

The U.S. role in the wold has not been as clear cut since. Failure in Somalia in the early 90's, limited success in Bosnia and Kosovo. The ultimate outcome of intervention in Afghanistan and Iraq is yet to be determined. Our lack of involvement in genocides in western Sudan has been criticized.

Don't get me wrong. I am blessed to be an American with Mayflower ancestors. It's just not as clear what God's politics are now.

See: What are We Fighting For? Western Civilization, American Identity, and U.S. Foreign Policy at the Foreign Policy Research Institute

Homosexuality - Gay Marriage

Jesus didn't have anything to say about homosexuality in particular. Some people say that his reference to sexual immorality (the King James translation uses fornication here) in Mark 7:20 refers to homosexuality.

Paul says in 1 Corinthians 6:9 that homosexuality will keep you from entering the kingdom of God.

Marriage:
I couldn't find anything in the bible specifically about marriage between a man and woman. There are several references to husbands and wives.
In Matthew 19, Jesus talking about divorce says "He which made them at the beginning made them male and female, And said, For this cause shall a man leave father and mother, and shall cleave to his wife: and they twain shall be one flesh?"

Over $74 million total was spent on campaigns both for and against California's Proposition 8 ballot initiative in 2008, which banned same-sex marriage and was passed 52% to 48%. It was subsequently overturned in court.

See Gay Marriage (Homosexuality)
and Divorce Rates

Environment:
Forty-five senators and 186 representatives in 2003 earned 80- to 100-percent approval ratings from the nation's three most influential Christian right advocacy groups -- the Christian Coalition, Eagle Forum, and Family Research Council. Many of those same lawmakers also got flunking grades - less than 10 percent, on average - from the League of Conservation Voters last year.
This is more difficult to explain based on the bible. Grist.org explains it as follows:
"Many Christian fundamentalists feel that concern for the future of our planet is irrelevant, because it has no future. They believe we are living in the End Time, when the son of God will return."

Socialism:
A lot of the argument by fundamentalist Christians, who in a large part are also politically conservative is that Communism was form of socialism and the USSR was also atheist therefore socialism is anti-christian.
Most people can see the problem with the logic here.

Jesus was not a sociologist or an economic reformer. Jesus never personally directed his followers to adopt a communal mode of life. He recognized the need for social justice and industrial fairness, but he offered no rules for their attainment.
Source: The Urantia Book - Fifth Epochal Revelation
However, the early church did in some cases seem to resort to communal living.
"All the believers were together and had everything in common. Selling their possessions and goods, they gave to anyone as he had need." (Acts 2:44-45)

Some passages which seem to promote socialism:
There was not a needy person among them, for as many as owned lands or houses sold them and brought the proceeds of what was sold. They laid it at the apostles' feet, and it was distributed to each as any had need. There was a Levite, a native of Cyprus, Joseph, to whom the apostles gave the name Barnabas (which means "son of encouragement"). He sold a field that belonged to him, then brought the money, and laid it at the apostles' feet.
(Acts 4:34-37)

The Parable of the Workers in the Field
'These men who were hired last worked only one hour,' they said, 'and you have made them equal to us who have borne the burden of the work and the heat of the day.' "But he answered one of them, 'Friend, I am not being unfair to you. Didn't you agree to work for a denarius? Take your pay and go. I want to give the man who was hired last the same as I gave you. Don't I have the right to do what I want with my own money? Or are you envious because I am generous?'
So the last will be first, and the first will be last. [Matt 20:1-16]

Parable of the Rich Fool - Luke 12:13-33
Sell your possessions and give to the poor. Provide purses for yourselves that will not wear out, a treasure in heaven that will not be exhausted, where no thief comes near and no moth destroys. 34 For where your treasure is, there your heart will be also.
Matthew 6:20 is similar.

In God's Politics: Why the Right Gets It Wrong and the Left Doesn't Get It, Jim Wallis' 2005 best selling book, he proposes a "prophetic politics" in which he appeals to the Old Testament prophets for inspiration and orientation. "It (prophetic politics) is traditional or conservative on issues of family values, sexual integrity and personal responsibility, while being very progressive populist or even radical on issues like poverty and racial justice."
See review at the Denver Seminary

Republicans and Democrats:
Republicans will claim they have higher moral standards than democrats because of things like stands on family values (e.g. their anti-gay positions) and sex and violence on TV.
Democrats will claim they have higher moral standards because of their stands on things like social justice, compassion, and the environment.
Republicans and fundamentalist Christians do have higher divorce rates.
See Divorce Statistics Republicans vs. Democrats
Moral-Values Politics: From Status Group Conflict to Institutionalized Party-Movement Electoral System (Seth Abrutyn) - Academia.edu

Links:
Social, Moral, Ethical Issues (Abortion, Homosexuality, Adultery, Women, ...)
Gay Marriage
What does the Bible say about marriage?
Biblical marriage: a bad source for debate
Fundamentalism
Books:
God's Politics, Jim Wallis
Partly Right: Learning from the Critics of Christianity, by Tony Campolo
Is Jesus a Republican or a Democrat?: And 14 Other Polarizing Issues by Tony Campolo