(Note: this is an abbreviated and therefore very simplified version of the history of the western church as I understand it. While I believe my general understanding to be correct, I am not an expert and may have inaccuracy in certain details.)
To begin our understanding of Christianity in America, we must go all the way back to the Roman Empire. For centuries, conquering states often pressured territories under their control to worship the Gods of the conquering state. In other words, there was one official religion of the state and if you lived in a land under its control, you were expected to practice this religion. Initially, Christianity was at odds with the Roman Empire for this reason. But around the forth and fifth centuries, Christianity became the official religion of the Roman Empire. Christianity began at a grassroots level. Over time, it developed more structure. The Roman Empire certainly solidified the institutionalization of Christianity. The legacy of this is what we know today as the Roman Catholic church.
During the 1500s, certain people believed Christianity was off-track. Martin Luther is the best known figure to push for change during this time. Though he advocated for reform, what happened was that churches in Germany broke off from (or were kicked out of) the Roman church, effectively inaugurating the Lutheran denomination and Protestantism (note the root word “protest”—they were protesting against certain aspects of Roman Catholicism). Lutheranism became the official religion of Germany.
There were others groups of Christians across Europe who had significant disagreements with the official religions of their states. Some of these groups believed that a person wasn’t a Christian just because he or she lived in a “Christian” land and practiced the rituals of Christianity. They believed an individual had to make their own commitment to follow Christ and had to follow his teachings as found in the Bible (rather than simply obey the directions of clergy) (see Anabaptists). Just as had been the case for centuries, the official religious leaders were closely connected to governing authorities. These beliefs were viewed as subversive and dangerous. Rebellion against the official church was viewed as rebellion against the state, and therefore was punished even by death.
It’s from this context that a number of religious groups from Europe sought refuge when the opportunity to move to “the New World” (America) opened up. Thus one of the important values held in America was the “freedom of religion”. However it is worth noting that the only “religions” they really had in mind were different denominations of Christianity. The idea behind the separation of church and state was that the state shouldn’t establish a single denomination as the official religion of the U.S., and should generally stay out of meddling in the affairs of churches. This wasn’t intended to mean that Christianity should never appear in public or government. Though a number of groups of Christians ventured to America to find the freedom to practice their denomination of Christianity, eventually nearly every denomination of Christianity arrived in America for one reason or another.
During the 18th century, two broad revivals were particularly influential on protestants in America, and led to a stream of Christianity known as Evangelicalism. Evangelical is a broad term generally describing protestants who emphasize the need for personal conversion, the authority of the Bible, the centrality of the death and resurrection of Jesus, and the desire to proselytize.
The Enlightenment beginning in mid-17th century Europe led to modernism, rationalism, and secularism. Some theologians sought to take modern approaches to Christianity and the Bible, leading to the development of liberal Christianity. Part of this included skepticism toward traditional understandings of the Bible and Christianity. The progress in science led some to declare that we no longer need God (see naturalism). This led in part to the rise of humanism which viewed the human being as the center of reality. (Most of these movements arose between the late 18th century and early 20th century.)
Concerned if not even alarmed by these movements in society, some evangelical Christians sought to reemphasize their beliefs in traditional understandings of the Bible and Christianity. This led to the rise of Christian fundamentalism in the early 20th century. A mark of fundamentalism has often been a belief the in need to fight against the perceived drift of the culture away from Christianity and/or perception of society’s growing hostility toward Christians. The rationalism of modernity also led to the popularity of apologetics among some Christians as a way to argue the reasonableness of their faith.
The relationship between evangelicalism and fundamentalism is complicated, in large part due to the fact that these are general labels rather than clearly defined denominations. However, fundamentalism could generally be viewed as a conservative side of broader evangelicalism. For example, while evangelicals generally believe in the authority of the Bible, fundamentalists emphasize literal interpretation and inerrancy. Though evangelicals cover a broad spectrum from progressive to fundamentalist, evangelical is often associated with its more conservative side, likely due in large part to the rise of the Christian right in the 1970s and 80s.
Traditionally, denominations represented the most significant divisions within Christianity. However, due to the movements which took place during the 19th and 20th centuries, the bigger divide in Protestantism now seems to be between those who lean more progressive and those who are more conservative in their approach to the Bible and theology. (Other significant differences include “high” church with liturgical worship verses casual worship, and the amount of emphasis on the work of the Holy Spirit.)