What is commonly referred to as the Establishment Clause appears in the first amendment to the U.S. Constitution. The entire amendment looks like this:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof; or abridging the freedom of speech, or of the press; or the right of the people peaceably to assemble, and to petition the Government for a redress of grievances.
The Establishment Clause is that first part: "Congress shall make no law respecting the establishment of religion." It basically states that Congress cannot establish a national religion or pass a law that essentially does the same.
The Establishment Clause stands in contrast to the situation in Great Britain at the time the Constitution was written. In the 16th century, English clergy and King Henry VIII broke with the Catholic Church, and the Church of England was established. What came to be known as Anglicanism was established as the national religion of England. Then, and still today, the king or queen of the United Kingdom is also the head of the Church of England.
Immediately after the Establishment Clause is the Free Exercise Clause — "or prohibiting the free exercise thereof" — which establishes the rule that the federal government cannot create laws that prohibit citizens from practicing the religion of their choice. Combined, the Establishment Clause and the Free Exercise Clause are the basis of the idea of the separation of church and state.