Congress shall make no law respecting an establishment of religion,

As the Supreme Court reads the Establishment Clause today, neither Congress nor the state governments can compel or encourage anyone toward any religious belief or practice. 

The most basic meaning of the clause is quite uncontroversial: there can be no "Church of the United States," as there is a Church of England.  Nor can there be a Church of South Dakota.  But if this were all the clause means, it would not mean much.  No Congress has ever tried to establish such a church, nor has any state has such an institution for a very long time.

This clause has been used extensively, however, to overturn any clear sign of public support for a particular religion or for religion in general, especially in the public schools.  However, it is sometimes ambiguous whether a state action is constitutes an establishment or not.  For example, a law requiring the posting of the Ten Commandments as part of a list of secular and sacred documents might be an attempt to teach a common history or might be an attempt to encourage Judeo-Christian religious beliefs.  To distinguish between permissible and impermissible laws, the court developed the Lemon test.

The Lemon Test: To survive scrutiny, a legislative act must:

1)    Have a secular purpose;

2)    Its primary effect must neither advance nor inhibit religion;

3)    It must avoid excessive entanglement between religion and government. 

This is probably more than what the clause was originally intended to mean.   To begin with, the language is odd.  It does not say merely that Congress cannot itself establish a National Religion, but that it can make no law "respecting" such an establishment.  This would seem to mean that Congress cannot forbid the state governments from establishing Churches, or interfere with such establishments several of which still existed at the time the amendment was adopted.