The fight over same-sex marriage is a relatively recent one. Initially, when activists tried to get it legalized state by state, there were some pretty significant setbacks. Notably in California, when a 2008 ballot measure once again restricted marriage to heterosexual couples only. The measure passed by a relatively narrow 52%-48% margin, having stopped (at least for the moment) gay marriage in its tracks in the nation's most populous and supposedly liberal state.
Still, Massachusetts and Connecticut had already legalized same-sex marriage, and soon after California's vote, Vermont and Iowa joined the ranks of those states allowing gay couples to marry. It looked like same-sex marriage supporters were in for a long slog.
In the other states, gay marriage had been established instead through the court system. Many activists believed that state judges, looking more dispassionately at state constitutions than many ordinary voters, would continue to be more likely to extend protection to same-sex marriage than state legislatures or the general electorate.
Basically, the idea boils down to "rights are only rights if you have a right to them." In other words, that they apply to everyone, and if they didn't, they'd be called something else. Wrongs, maybe.
Opponents of gay marriage decry these tactics as "judicial activism," the antidemocratic practice of appointed judges deciding political matters that should rightly be left to elected officials or the voters themselves to decide. Apparently people don't like it when judges…judge.
On June 26, 2015, the Supreme Court handed down a decision on Obergefell et al. v. Hodges, Director, Ohio Department of Health et al. (known as Obergefell v. Hodges for short). Surprisingly, this had nothing to do with "Obergefell" being a fake name. Nope, this was the gay marriage decision that your uncle keeps posting on Facebook about.
This Supreme Court case was about whether or not same-sex couples had the right to get married under the 14th Amendment to the Constitution.
Incidentally, this is no longer a debate.
Because of how the government works, the Supreme Court answered, 5-4, that yes, same-sex couples can be married, and it is in fact unconstitutional to deny them that right. We're done. Everyone can go home now, unless you have an awesome wedding to go to. There are more of those nowadays.
So what was the question to begin with?
In the majority opinion, Justice Anthony Kennedy wrote
Under the Due Process Clause of the Fourteenth Amendment, no State shall 'deprive any person of life, liberty, or property, without due process of law.' The fundamental liberties protected by this Clause include most of the rights enumerated in the Bill of Rights.
So the question Kennedy leaves open here is this: Is marriage covered under "life, liberty, or property"? Or more to the point, does marriage affect those three things in any way? (Hint, it totally does.)
He went on to say that "the Court has long held the right to marry is protected by the Constitution" (source). He cited Loving v. Virginia, the case that legalized interracial marriages (yep, those used to be illegal, too), as a precedent. He listed a few others, but Loving v. Virginia draws the most explicit connection in the civil rights sphere.
While the four "liberal" justices joined Kennedy's majority decision without comment, the four "conservatives" explicitly challenged the Constitutionality of the ruling, arguing that marriage as a tradition pre-dates the Constitution.
What does this mean for same-sex couples? They get the same right to marry as opposite-sex couples. People enter marriage by obtaining a license from local authorities, and—in about half of all marriages these days—they later end it by securing a divorce decree from a local civil court.
In between, the United States Government Accounting Office has identified more than a thousand federal benefits and responsibilities (such as homebuyer programs and educational assistance) affected by marital status. Translation: there are thousands of laws touching on marriage.
The Fourteenth Amendment's guarantee of equal protection under the laws has been interpreted that all people, regardless of sexual orientation, should be protected in their right to marry, which, in turn, has given them loads of other rights, too.
What does this mean for opposite-sex couples, or single people who don't want to get married? Absolutely nothing. Throw a party. Throw a fit. Have a wedding for penguins. Boycott weddings all together. The only thing you can't do is stop people from getting married—because that's now illegal.