The First Amendment to the United States Constitution, commonly known as part of the Bill of Rights, states in part that "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof..." Much later, the Civil Rights Act of 1964 made broad strokes to prohibit discrimination based on religion or association with those of any religion. A key court case that helped further the concept of separation of church and state was the 1947 Supreme Court Case known as Everson v. Board of Education, in which it was considered a misappropriation of funds to use school moneys to transport children to religious events. In 1962, this separation extend further by prohibiting group prayer in school, especially as composed or organized by teachers or administrators.
Despite these laws, the separation of church and state remains a hot button issue, with many other legal actions pending against various government or public agencies that would seem by their actions to endorse a specific religion. In the main, much of the way the US government works is not specifically tied to any church. Political candidates have certainly used their own religious status to appeal to those of similar religious views, however.
It’s hard to argue that the separation is complete in the US. The postal service, for instance, doesn’t deliver mail on Sunday, which makes no sense to those who celebrate the Jewish Sabbath, occurring on Saturday. Similarly, Christmas is a federal holiday, but Hannukah is not, nor Rosh Hashanah. Government offices don’t observe the fasting practices of Ramadan.
US currency and the Pledge of Allegiance (not taken by Jehovah’s Witnesses) also reference God, although they don't refer to a specific religion. There has been recent movement to strike the "under God" reference from the Pledge of Allegiance, but any pledge to a flag may be construed by some religious groups as placing a nation above God.
As for prayer in public schools, the idea of separation of church and state gets muddied and laws become very difficult to interpret. In some schools, even calling for a moment of silence is considered as crossing into dangerous territory, yet many US presidents sign off on presidential addresses by entreating God to bless America. Further, some people will not vote for presidential (or any political) candidates who do not go to church or who have a religion that they feel conflicts their own. So religion does have some influence on who runs the US, and who stands in office.
In courts across the nation, people may swear that their testimony is true on a bible, and there are several states where political officials take oaths that include the words God. Chaplains are employed by the military, and military officers may pray at meals, though this is voluntary. It’s fairly clear from these examples that separation between church and state isn’t whole and entire. It’s also clear that the framers of the US Constitution built specific moral codes into the work that suggest a belief in God, and though the intent may have been that one church should not have control over the government, it may not have been to strike out belief in God or recognize commonalities among many Christian groups, making the US a "God fearing nation" or marking currency with statements like "in God we trust."
The question then of whether separation of church and state exists in the US is one under constant scrutiny. There are those who argue that the state minimally involves reference to any "church," and others who argue that a Christian religious viewpoint remains a constant influence over the government and is preferenced by the state. Each US citizen must ask if this matters, and if so, how much; if the practice of a specific religion within state context impinges on the rights of others or insult their free practice of religion; and how much people have divided church and state and if more or less division purposeful or useful.
