DOMA is a 1996 law passed by Congress, and signed into law by President Bill Clinton, that forbids the federal government from recognizing same-sex marriages.
In a heterosexual marriage, if you get married in Massachusetts and move to Texas, nothing changes; your marriage is still official. DOMA meant same-sex marriages were only valid in states that allowed them.
It was implicit federal approval of marriage discrimination. Bill Clinton and others who created the bill have since come out against DOMA, and in 2013 the Supreme Court ruled that section 3 of the act - the one that allowed states to refuse to acknowledge same-sex marriages - was unconstitutional. This decision probably sparked all the interest in DOMA during 2013.