The First Amendment to the US Constitution outlines the legal relationship between government and religion. But the First Amendment deals with broad principles, and leaves much to the interpretation of the courts (this does not lead to an anything-goes free for all: the First Amendments principles are pretty clear, but the courts have had to work out how to apply them to specific situations).
One pivotal First Amendment case was Lemon v. Kurtzman. The details of the case can easily be found online or in any library, so I will not delve into them here. But one result of the case was the articulation of the so-called “Lemon Test,” which attempted to more specifically delineate the role of government with respect to religion. The Lemon Test has three components or “prongs;” government action does not run afoul of the First Amendment if it remains within the parameters of these three criteria:

The government’s action must have a secular legislative purpose;
The government’s action must not have the primary effect of either advancing or prohibiting religion;
The government’s action must not result in an excessive entanglement in religion [and clearly there is a subjective element here].

This is not to say that the Court’s ruling is perfect, but it does give us a starting point, and it is important to know about the Lemon Test when we discuss Church/State issues.