After the election of Thomas Jefferson as the third president of the United States in 1801, the Danbury Baptist Association in the State of Connecticut wrote a letter to Jefferson in which they explained that as a religious minority, they feared that the government of Connecticut would legislate on the side of the religious majority.

[...] Sir, our constitution of government is not specific. Our ancient charter, together with the laws made coincident therewith, were adapted as the basis of our government at the time of our revolution.

And such has been our laws and usages, and such still are, that Religion is considered as the first object of Legislation, and therefore what religious privileges we enjoy (as a minor part of the State) we enjoy as favors granted, and not as inalienable rights. And these favors we receive at the expense of such degrading acknowledgments, as are inconsistent with the rights of freemen.

Their fear was that the wording of the Establishment and Free Exercise Clauses of the First Amendment essentially outlined what the government couldn't do in broad terms in respect to religious freedom - but didn't outline what the natural religious rights of man are. What the government couldn't do in regard to religion was something they feared could be chipped away at by the whims of the majority.

Jefferson's response gave birth to the phrase "separation of church and state":

Believing with you that religion is a matter which lies solely between man and his God, that he owes account to none other for his faith or his worship, that the legislative powers of government reach actions only, and not opinions, I contemplate with sovereign reverence that act of the whole American people which declared that their legislature should "make no law respecting an establishment of religion, or prohibiting the free exercise thereof," thus building a wall of separation between church and State. Adhering to this expression of the supreme will of the nation in behalf of the rights of conscience, I shall see with sincere satisfaction the progress of those sentiments which tend to restore to man all his natural rights, convinced he has no natural right in opposition to his social duties.

Now Jefferson's declaration of a "separation of church and state" didn't get much fanfare until the Supreme Court took on one of its first religious freedom cases, Reynolds v. United States in 1879.

The case involved a Mormon charged with bigamy who argued that his First Amendment right of free exercise of religion was being infringed upon. The Court invoked the metaphor of the "wall of separation" in its decision that laws prohibiting polygamy/bigamy were constitutional. While the First Amendment protects the free exercise of religion, the Court reasoned that the government could legislate against religious practices that had a serious negative societal effect. The "wall of separation" metaphor goes the other way, essentially: if the government were to allow Mormons and no one else to practice polygamy, then that would in effect be a tacit endorsement of a religious freedom.

But the case that really ingrained the concept of a "wall of separation" into constitutional jurisprudence was Everson v. Board of Education in 1947. At issue was a New Jersey law that provided financial assistance to all public and private schools to use for transportation expenses. The plaintiffs held that giving tax dollars to private religious schools was a violation of the Establishment Clause.

Justice Hugo Black's famous majority opinion held that the New Jersey law was constitutional because the reimbursements were offered to all students:

"The 'establishment of religion' clause of the First Amendment means at least this: Neither a state nor the Federal Government can set up a church. Neither can pass laws which aid one religion, aid all religions or prefer one religion over another. Neither can force nor influence a person to go to or to remain away from church against his will or force him to profess a belief or disbelief in any religion. No person can be punished for entertaining or professing religious beliefs or disbeliefs, for church attendance or non-attendance. No tax in any amount, large or small, can be levied to support any religious activities or institutions, whatever they may be called, or whatever form they may adopt to teach or practice religion. Neither a state nor the Federal Government can, openly or secretly, participate in the affairs of any religious organizations or groups and vice versa. In the words of Jefferson, the clause against establishment of religion by law was intended to erect 'a wall of separation between Church and State.'"

Black's position on First Amendment religious rights has since been a cornerstone of textualist/originalist constitutional interpretation, though the "wall of separation" metaphor is invoked by pragmatists as well. The more conservative wing of the Supreme Court - generally textualists and originalists - tend to view the wall as a barrier that protects religion from government, while the more liberal wing of the Court - pragmatists and living constitutionalists - tend to view the wall as a barrier that protects government from religion.

Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof...

In and of itself, this first part of the First Amendment just talks about the federal government: Congress may not establish one religion above the others, nor prohibit anyone from exercising his religion. Some states still did have established churches in the 1790's; the last state to disestablish its church was Massachusetts, in 1833. It was the Fourteenth Amendment several decades later which incorporated this against the states together with the rest of the Bill of Rights.

This's been popularly referred to as "separation of church and state." The phrase comes from Thomas Jefferson's letter to the Danbury Baptist Association in which he said the First Amendment had built "a wall of separation between Church & State." While the phrase never appears in the Constitution, it's a generally-correct way of describing what the First Amendment does say.