Religious exemption is a right in Federal law (if not in the First Amendment itself) to not be required to do certain things which go against one's religious viewpoints.
For example, a person who is a Seventh Day Adventist (and therefore, observes Saturday as the Sabbath) may request an exemption to not be required to work on Saturdays. The employee may offer an alternative, such as to work on Sunday instead.
Under current Federal law, no specific membership in, or adherence to, any organized religion is required to claim an exemption: the requestor is only required to have a "sincerely held belief" in order to do so. However, it is not automatic if an employer can show that granting the exemption would create an "undue hardship". In the example above, if the business is closed on Sunday, the employer might be able to prove an "undue hardship" and therefore an exemption would not be granted.
The issue has surfaced due to certain requirements regarding mandatory vaccination. Numerous passages in the Bible supporting objecting to ingesting unclean substances -- i.e., certain vaccines -- particularly in the letters from Saint Paul in the New Testament.