For a good overview of the decision itself and the legal impact, visit our friend's blog over at Finn & Eaton: Obergefell v. Hodges: Supreme Court Rules on Same-Sex Marriage, Part 2.
For now we're posting the tl;dr version of the majority and dissenting opinions:
Majority Opinion written by Justice Kennedy and joined by Ginsburg, Breyer, Sotomayor and Kagan
tl;dr: Marriage is a fundamental right under 14th amendment due process and extends to same-sex couples because there is no rationale to deny benefits based on the societal reasons for promoting marriage. This was not based on a typical equal protection analysis, but rather focused on the nature of marriage and why it's important in today's society and therefore represents an inalienable liberty.
Justice Roberts' dissent joined by Scalia and Thomas
tl;dr: Extending due process rights to marriage = judicial law-making because the history of marriage is clear and the legislative process was doing just fine. Criticizes the majority for ignoring typical equal protection analysis.
Justice Scalia's dissent joined by Thomas
tl;dr: Scalia really wants you to know that he agrees with Roberts but wants to drop the pretense that he "respectfully dissents". Also, Californians aren't real westerners and hippies have more fun (seriously, he said that).
Justice Thomas' dissent joined by Scalia
tl;dr: You think you know what "liberty" is? Well, do I have an irrelevant history lesson for you. It basically involves proving that liberty and dignity can't be given to you by the government so people shouldn't care if the government endorses slavery or traditional marriage. Change comes from within, man, so don't look to them for help.
Justice Alito's dissent joined by Scalia and Thomas
tl;dr: Same sex marriage might have dire consequences... none of which he's willing to describe in detail other than to imply that religious freedom and democracy itself will be damaged by this decision.