In monogamy a person has only one spouse at a time (as opposed to polygamy).
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.
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 unlike some other species, not all members of Homo sapiens are naturally monogamous, and the 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.
Monogamy is also occasionally used to mean having only one sexual partner during an entire lifetime.