The Supreme Court has announced its opinion in Obergefell v. Hodges, observing that "As the State itself makes marriage all the more precious by the significance it attaches to it, exclusion from that status has the effect of teaching that gays and lesbians are unequal in important respects. It demeans gays and lesbians for the State to lock them out of a central institution of the Nation’s society." The Court holds that "the right to marry is a fundamental right inherent in the liberty of the person, and under the Due Process and Equal Protection Clauses of the Fourteenth Amendment couples of the same-sex may not be deprived of that right and that liberty....[S]ame-sex couples may exercise the fundamental right to marry."
My article on the immigration effects of same-sex marriage after the Court's decision in U.S. v. Windsor sets out the different ways in which a same-sex couple could seek immigration benefits after the Court overturned the federal definition of marriage in the Defense of Marriage Act (DOMA) as solely heterosexual, and allowed immigration agencies to recognize same-sex marriages and give federal benefits to same-sex married couples. Sidra Vitale, Immigration Equality: How DOMA's Repeal Affected Immigration, 48 New Eng. L. Rev On Remand 7 (2013).
These effects are unchanged by this decision. What is changed is that immigrant couples won't need to go to another state to be lawfully married and then return home to seek their immigration benefits. What is also changed is the panoply of state rights and benefits associated with marriage that have been denied to same-sex couples until now. The DOMA decision opened up various federal rights to same-sex married couples (including immigration), this decision brings true equality under the law by ensuring recognition at the state level as well.