In monogamy (Greek: monos = single/only and gamos = marriage) a person has only one spouse at a time (as opposed to polygamy). Monogamy is also usually used to mean having only one sexual partner during an entire lifetime. In the animal world, monogamy is the practice of remaining with one partner for sexual reproduction and the raising of young.
The practice of restricting sexual contact to a single partner (married or not) for a limited period of time, then ending that relationship before beginning another (though in practice there may be a brief overlapping time-period) is referred to as serial monogamy (as opposed to polyamory, swinging, etc.).
Historically, monogamy was much less practised than polygamy (specifically polygyny). Mostly because of European expansion, monogamy is more popular than it was ever before. See article about polygamy for details.
Polygamy as an institution continues in much of the developing world. It should be noted, however, that even where polygyny is allowed, it is less than commonly practised, as few men in such communities have the financial means at hand to support additional wives. It is usually observed in groups of people that have recently experienced war or famine - disasters which typically kill proportionally more men that women. The Anglican Church in Kenya for example, has semi-officially adapted a positive stance on polygamy, largely because of deficit of males in that country due to decades of war.
Polyandry, or the practice of women having more than one male spouse, is traditionally a rarer phenomenon than polygyny. The most famous example of polyandry, in Hindu culture, for example, occurs in the Mahabharata where the Pandavas are married to one common wife, Draupadi. Today it is almost exclusively observed in the Toda tribe of India, where it is sometimes the custom for several brothers to have one wife. In this context, the practice is intended to keep land - a precious resource in a populous country like India - within the family.
Although modern groups that advocate polyamorous relationships attempt to construct historical or archaeological evidence as favouring these types of relationships as "natural", it is impossible to portray human relationships as simplistically as this. Humanity's closest relatives, the bonobo and the common chimpanzee display very different types of sexual behaviour - chimpanzees favour fairly rigid hierarchical relationships while bonobos are openly promiscuous. Other close human relatives such as marmosets and gibbons are more or less monogamous in their habits. It should also be noted that the Neandertal lived in small groups revolving around a single breeding couple.
While most pre-modern societies exhibited varying degrees of polygamy, in most instances, pair-bonding was more commonplace than not. It is interesting to observe that even in cultures that permit polygamy, its practice may nevertheless be discouraged. The Islamic Qu'ran, for example, suggests men restrict themselves to one wife: "If you have more than one wife, you will never be able to treat them equitably...and if you cannot treat them equitably [then you should not engage in the practice at all]."
Note that the term "monogamy" is also used to mean confining a sexual relationship to one other person even in the absence of a legal status of marriage (for example, an unmarried heterosexual couple or a homosexual couple in a jurisdiction that does not recognize marriage between homosexual persons.) Monogamy in this sense is recommended by health professionals discussing safer sex practices.
Some argue that polyfidelity, which is restricting oneself to a group of people, would provide the same protection if each person who has sex follows the same rules, stays in the group, and the only way to join is a negative test for STDs. The lack of completely reliable tests for STDs -- false negatives can occur and some STDs, such as HIV, are not detectable by blood test for up to six months after exposure -- suggests, however, that even in the absence of cheating, the risk would increase for each participant.
Note also that existence of a legally monogamous relationship (marriage) is no guarantee of a monogamous one in fact. Some societies have formally or semiformally recognized that married persons may have other sexual partners outside of the marriage relationship, while in societies that do not condone this practice it is nevertheless not unusual.
- "The Virtues of Promiscuity" by Sally Lehrman http://www.alternet.org/story.html?StoryID=13648 - on studies showing social and genetic benefits of promiscuity