Bigamy - marrying another person while still being married - is illegal in the United States. You can receive fines and / or prison time, depending on the state. For example, in Utah, Bigamy is a "Felony of Third Degree", meaning you can have between 2 and 10 years in prison (see Sec. 12.34.). Child bigamy second degree, and so is 2 to 20 years (see Sec. 12.33.). It is also enforced.
Polygamy is different to Bigamy - it is broader. It includes bigamy, and is not used in legal context. Polygamy also doesn't have to be illegal. You can hold a religious ceremony and declare yourself married to 3, 5, 10 - as many people as you want. As long as you only legally marry one of them, you've not committed a crime. This is because of the difference between a legal marriage and a religious marriage in the US. However, living with them, and committing adultery is illegal in 21 states, but it often seems to be that:
no prosecution for adultery shall be commenced except upon complaint of the husband or wife
Despite that, This site says that someone is having to argue for them to enforce a law (which seems a little strange), and the post suggests he might not win (saying "could be prosecuted").
US law was also based on English Law (3rd Paragraph):
The principle that a person could only be married singly, not plurally, existed since the times of King James I of England in English law, upon which United States law was based.
The 1878 court case Reynolds v. United States was an important test case.