Polygamy, the "state or practice of having more than one spouse simultaneously," was outlawed by the federal government in 1862 [Black's Law Dictionary]. Still, polygamy exists in the United States, though usually secretly and with great stigma attached. One can also find polygamy in popular culture, such as in the television show "Big Love." In the U.S., polygamy is frequently associated with Mormonism, although the Mormon church condemns the practice, which is also called plural marriage. An estimated 30,000 to 60,000 polygamists live in Utah and the surrounding states, most of them made up of fundamentalist Mormons [ref]. For more information about polygamy, check out How Polygamy Works.
Civil Unions, Domestic Partnerships, Gay Marriage
A civil union is a legal entity that offers same-sex couples the same legal rights and benefits as marriage. Civil union laws have been proposed in various state legislatures, but so far they are only legal in Vermont, New Jersey and Connecticut [ref].
California allows domestic partnerships for same-sex couples and for opposite-sex couples in which one partner is at least 62 years old, and many employers and universities offer domestic partnership benefits as well. Like civil unions, domestic partnerships offer many of the same legal benefits that opposite-sex married couples receive, including inheritance and hospital visitation rights. Besides California, domestic partnerships are allowed in New York City, Maine and the District of Columbia.
Hawaii allows "reciprocal beneficiary relationships," which are registered through the state's Department of Health [ref]. Reciprocal beneficiary relationships are only available to people who are at least 18 years old and prohibited from marrying by state law, which includes same-sex couples but can also mean a brother and sister or aunt and nephew [ref]. Two people who enter a reciprocal beneficiary relationship aren't automatically considered a couple -- it simply means that they gain many of the legal rights afforded to married couples, which, for example, could be important to a brother and sister who are supporting each other financially.
The Defense of Marriage Act of 1996 defined marriage as "only a legal union between one man and one woman as husband and wife" [ref]. This definition applies for federal purposes only, and states are free to create their own laws with regards to marriage. However, states are not legally required to recognize same-sex marriages conducted in other states that allow them. Currently, Massachusetts is the only state in which gay marriage is legal. Some states have explicitly outlawed gay marriage, while a constitutional amendment to ban gay marriage has failed more than once to pass in the United States Senate [ref].
Common Law Marriage
In order for a couple to be considered to have a common law marriage, they must live together, agree that they are married and present themselves as a married couple, such as by changing their last names and filing joint tax returns. If you've heard that any couple that resides together for seven consecutive years is automatically considered common law husband and wife, that's not true. The amount of time that a couple has to live together is not defined in any state, but if a couple satisfies many of the conditions for common law marriage and lives in a common law marriage state but does not want to be married, they must somehow make that intention clear.