I have actually taken an anthropology course entitled "Magic, Religion and Science," which did a great job of introducing primary topics in the anthropology of religion.
It is important to first understand that today's meaning of "religion" is a very recent term in human history. Romans used early versions of this term, including "religio," to refer to various patterns of worship. Early Christians used the same meaning, but it was not until the 1600s that Europeans used forms of the word "religion" to refer to specific set of beliefs. Not coincidentally, this was the same of European expansion. Europeans of this era were most familiar with Christianity and Judaism, and so they used these faiths as models for what a religion is (and is not). There were actually specific traits a belief system & its people were supposed to have in order to be classified as a religion. Although the topic and hand is not the history of religion, it is important to consider these early standards for religion as we expolore religions and belief systems around the world.
Let's consider Japan: In a 2000 survey on religion and religious practices in Japan, 52% of the hundreds of Japanese who participated indicated that they are NOT religious. However, 36% of participants reported that they are Buddhist (one of the country's primary belief systems), 11% stated that they belong to Shinto faith (the indigenous, nature-based faith said to be as old as the Japanese people), and 11% stated that they associate themselves with Christianity. Now, with such a tremendous report of religious affiliation, why do over half of the Japanese report that they are not religious?
The Japanaese language actually did not even have a term for religion until the Europeans came along (and I mean this quite literally). The Japanese literally created a term to adopt the European idea of religion. I stated earlier that the European definition consisted of specific traits which make a person religious. These are: 1), Religions were supposed to have one central text, such as the Bible or Torah, 2), people were supposed to "belong" to one religion only (the practice of exclusivity), and 3), religion was supposed to make up a distinct social doman, separate from other domains such as economics and government (although many would argue that some European countries have NOT separated church and state).
Many Japanese do not practice exclusivity; a person may visit Shinto shrines, have a Christian wedding ceremony, and then have a Buddhist-style funeral ceremony. Also, the Japanese (in addition to MANY other cultures around the world) do not always separate religious/spiritual beliefs from everyday life. In fact, of the estimated 20% of Japanese who belong to a "Japanese New Religion" such as Mahikari, many admit that they use faith to seek answers and/or advice in regards to jobs, the economy and other matters. As we begin to consider these and other details of Japanese religious life, it becomes easier to see why so many Japanese do not consider themselves religious; the Euro & western notions of what a religion is (and is not) does not necessarily coincide with Japanese practices. As a result, they do not assign themselves the religious label.
Moving away from Japanese faith, one can consider the European definitions of religion to understand why certain groups have been targeted by missionaries. Considering that Europeans of centuries past required that a religion have a central text in order to actually BE a religion, there is a seemingly innumerable number of cultures who appeared to "have no religion" at all due to their lack of central text. This perceived lack of religion has made it easy for missionaries to justify their impositions on the cultures/groups they target. In fact, in the early days of European expansion, it was not uncommon for Europeans to say to practicioners of indigenous, non-Christian faiths that they lacked religion entirely (this still happens today in some places). Think about the wide variety of indigenous cultures with their own belief system; from the !Kung of southern Africa to the Sambia of Papaua New Guinea, there are SO many groups who have a belief system, but do not fit with European models of what makes up a religion.
Okay, so we're all different. So, what? What's so important about all of this?
Actually, some problems HAVE developed as a result of rigid standards on religion. Indonesia, for example, has a government which only recognizes six religions: Islam, Protestantism, Catholicism, Buddhism and Confucianism. Another thing about Indonesia is that there are several indigenous cultures which practice their own belief systems. Some of these cultures have resorted to manipulating the structure of their belief systems so that they appear to fit into one specific religion. For example, one indigenous culture borrowed some statements and practices from the Qur'an and publicly stated that they follow the Qur'an simply so that their group could be recognized by the government has having some acceptable religious grounds.
Ethical issues also arise due to the emergence of new faiths. Mary Baker Eddy founded the Christian Science Church, which professes that healing occurs through prayer, or by reading from one of the religion's two central texts: The Bible, and Eddy's book, entitled "Science and Health." This is the religion which is known for rejecting medical attention. One famous case is that of David and Ginger Twitchell, practitioners of this religion who were convicted of involuntary manslaughter in 1990 when their two-year-old son died due to "curable ailments" after they refused to allow him to be treated by physicians. How should governing parties respond to their religious justification?
These and other issues are prominent in the anthropology of religion because they require a certain insight, which can be reached through a cultural relativist approach to the study of religion.