"No Religious Test"

What does it mean when folks say that there should be no religious test for public office?

In regard to the courts, at least, I believe that legal analysis should have no express or implicit religious basis, and there should be no discrimination in favor of certain religions over others, or over the areligious.

However, other than that, religious feeling should not be outlawed among public officials. Why should it?

Indeed, there should be no religious test -- which to me means that if one is religious, and even extremely so, one should not be prevented from holding office.