The Defense of Marriage Act (DOMA) is a law that was passed in 1996 as a way of preventing same-sex marriages from being recognized on the federal level. The law also sought to ensure that states that did not want to recognize gay marriages would not have to do so.
One main aspect of the DOMA was that it defined marriage as the union of one man and one woman for the purposes of the federal government. This meant that same-sex couples could not have any of the rights of married people. For example, same-sex couples do not have the right to file joint tax returns even if they are legally married in the state where they live.
A second main aspect of the law is that it frees states from the obligation of honoring same-sex marriages performed in other states. In general, states simply accept marriages from other states. You do not have to get remarried if you move from state to state. However, the DOMA explicitly says that states do not have to honor same-sex marriages from other states.