Because one has not been put forth here yet, and because I feel it is an important part of thinking about religion, I will put forth an anthropological definition/anthropological thoughts about religion.
First, however, some things on which it is based:
- One cannot prove the existence of a supernatural power. One cannot prove the non-existence of a supernatural power. Thus, logically (and scientifically), nothing can be said about the existence of a supernatural power.
- Anthropology, as a science, thus makes no judgements, theological or otherwise, about the validity of any religion, because nothing can be scientifically proven about them.
- What anthropology does, however, is look at why religion exists, how it came about, how it changes; and how it affects society, culture, and people, and vice-versa.
So, with that in mind: science gives us answers to a lot of things. Why the sun rises and sets, why blood transfusions of certain different blood types will kill a person, why people get diseases and why they get better or not, and why sometimes it rains and sometimes it doesn't. With this knowledge comes a certain amount of control over some of these things.
However, there are some questions that science can't answer. What happens to us after we die? Why are we here? What is the meaning of life? What was there before the big bang? How should I treat my neighbor? These are some pretty serious questions, especially the first three. So serious, in fact, that not knowing the answers causes anxiety in people. Anxiety intense enough that some of them start looking elsewhere for answers.
Religion is one source of answers. However, religion is a matter of faith, as in the basis for its answers do not come from empirical evidence. Thus religion, by its very nature, does not need to prove itself to anyone. Indeed, if it did, it really wouldn't be a religion anymore. It would be a fact.
It's pretty obvious how religion arose. Imagine living 10,000 years ago, with no knowledge of any kind that would tell you why someone sickened and died; why lightning struck nearby; why it didn't rain for months on end and you and your family had to move or starve; or even how a woman gets pregnant. That's a lot of unanswered questions and very little control over your life, and people wanted answers. There were no logical ones at the time, so they came up with supernatural explanations based on faith. And these answers gave them a feeling of control, whether real or imagined.
Over time, science gives us more and more answers. But we still don't know everything, and some people still want a comforting answer, and thus religion still exists today.
Way back when, a religion was often integrated directly into a culture, which more often than not was the only society in an area. You didn't have much choice in the way of your beliefs. As the earth's population grew, different cultures came into contact with each other, with mixed results. Some societies started to incorporate multiple cultures, while others sought to force their culture and/or religion on the "outsiders". Religion created, and still creates, a sense of solidarity amongst its followers. It gives you a feeling of belonging, a sense that you have a support group. In doing so, it creates another group, the outsiders.
So now, regardless of what culture you're born into, you are exposed to numerous other cultures and religions within your society. And if you are in a society that allows it, you can now choose the aspects of your own individual culture and religion (or un-religion).
Religion can obviously be used as a tool for good or ill. It can be used as a means of moral engineering. Many people do not realize that religion is not the only source of morals and ethics. Many others hold that they are right in their beliefs, and everyone else's beliefs are wrong, and this has caused all sorts of problems, and continues to do so.
So, to wrap it up, religion is any set of beliefs which is based entirely on faith and not on empirical evidence (which means that one person can have their own religion, although this tends to get called a cult until it builds a sufficient following). It functions to help the mental health of the believer, provide a sense of solidarity with those who share that set of beliefs, and can be integrated into a culture and possibly a society. Religion is actually at one end of a spectrum of belief systems, which also includes sacrifice, witchcraft, sorcery, and magic. See also the life cycle of a religion.