Monogamy is the practice of having a relationship with only one sexual partner at a time. Some would say that it applies only to marriage but since there is no 'mono' version of polyamory the term tends to be used more loosely to include unmarried relationships as well. An even more strict definition of monogamy is that is it the practice of marrying only once in a lifetime, though this is rarely applied to human behaviour (as opposed to deuterogamy).
Given the loose way the word is used, one also finds people using the term in the context of a married person who is also having sex outside the marriage in an affair. They are described as monogamous because that is the relationship their partner believes they have, even though the person is clearly not having only one sexual relationship at a time.
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.).
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 is believed that the Neanderthal 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.
Monogamy is often 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 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 semi-formally recognised 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 word 'monogamy' derives from Greek: monos = single/only and gamos = marriage. Hence the strict definition of a person who has only one spouse at a time.