What are the constitutional issues regarding things like prayer in school?

The basic issue in these cases has to do the the
establishment clause in the First Amendment.  This is the part of the amendment that
says that the government cannot make laws "respecting an establishment of
religion."

At its most basic, this means that the
government cannot set up an official state religion.  The clause is also taken to mean
more generally that the government cannot promote religion.  The Supreme Court has set
up a three part test known as the "Lemon Test" to determine when the government is doing
too much to promote religion.  As the link below says, the Lemon Test says that a
governmental action is acceptable if it

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(1) has a secular, or non-religious, purpose;
(2) has a main effect that neither advances nor restricts religion; and (3) does not
foster excessive entanglement, or mixing, between religion and
government.

On issues of
prayer in school, the courts have had to prevent violation of this test while also
preventing the violation of student rights to free speech.  The general rule has been
that any prayers at school events have to truly be spontaneously done by students.  The
prayers cannot be approved of by the school or in any way promoted by the
school.