Fourteen same-sex couples, and two men whose same-sex partners had passed away, filed suits in their home states of Michigan, Kentucky, Ohio and Tennessee. At the time, all four of these states defined marriage as a union between one man and one woman. The petitioners said that this narrow definition violated the 14th Amendment because it denied them the right to get married (or have the marriages they'd received in other states legally recognized at home).
After the trial courts in each state sided with the plaintiffs, the rulings were appealed in the U.S. Court of Appeals for the Sixth Circuit, where the decisions were reversed, and it was ruled that state bans did not in fact violate the couples' 14th Amendment rights to equal protection and due process. This split led to the Supreme Court review [sources: Supreme Court, Oyez].
Ginsburg voted with the majority on this one, in the 5-4 ruling that held that same-sex marriage bans are indeed violations of the 14th Amendment's Due Process and Equal Protection Clauses. "The Constitution promises liberty to all within its reach," Justice Anthony Kennedy wrote in the majority opinion, "a liberty that includes certain specific rights that allow persons, within a lawful realm, to define and express their identity" [sources: Supreme Court].
This landmark civil rights case legalized same-sex marriage across the United States, giving hundreds of thousands of LGBTQ Americans the same rights and protections guaranteed to heterosexual couples by both the Due Process Clause and the Equal Protection Clause of the 14th Amendment to the United States Constitution.