Free Exercise Clause refers to the section of the First Amendment italicized here:
Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof...
The Free Exercise Clause reserves the right of American citizens to accept any religious belief and engage in religious rituals. Free-exercise clauses of state constitutions which protected religious “[o]pinion, expression of opinion, and practice were all expressly protected” by the Free Exercise Clause. The Clause protects not just religious beliefs but actions made on behalf of those beliefs. More importantly, the wording of state constitutions suggest that “free exercise envisions religiously compelled exemptions from at least some generally applicable laws.” The Free Exercise Clause not only protects religious belief and expression; it also seems to allow for violation of laws, as long as that violation is made for religious reasons. In the terms of economic theory, the Free Exercise Clause promotes a free religious market by precluding taxation of religious activities by minority sects.
Constitutional scholars and even Supreme Court opinions have contended that the two religion clauses are in conflict. E.g., Thomas v. Review Board, 450 U.S. 707 (1981). As mentioned previously, the Free Exercise Clause implies special accommodation of religious ideas and actions, even to the point of exemptions to generally applicable laws. Such a special benefit seems to violate the neutrality between “religion and non-religion” mandated by the Establishment Clause. McConnell explains:
If there is a constitutional requirement for accommodation of religious conduct, it will most likely be found in the Free Exercise Clause. Some say, though, that it is a violation of the Establishment Clause for the government to give any special benefit or recognition of religion. In that case, we have a First Amendment in conflict with itself—the Establishment Clause forbidding what the Free Exercise Clause requires.
Historically, the Supreme Court has been inconsistent in dealing with this problem. At various times, the Court has either applied a broad or narrow application of the clause.
When the First Amendment was drafted, it applied only to the U.S. Congress. As such, state and local governments could abridge the Free Exercise Clause as long as there was no similar provision in the state constitution. In 1940, the Supreme Court held in Cantwell v. Connecticut that, due to the Fourteenth Amendment, the Free Exercise Clause is enforceable against state and local governments (this act of using the Fourteenth Amendment as the vehicle through which the Court applies the Bill of Rights to the states is also known as the Incorporation Doctrine).
 Michael McConnell, Religion and the Constitution (2002), pg. 105.
 Id. at 107.
 Richard Posner and Michael McConnell, "An Economic Approach to Issues of Religious Freedom," 56 University of Chicago Law Review 1 (1989).
 McConnell, note 1 above, at 102.