The Gist: Before this decision, some states allowed same-sex marriages, while others banned it, defining marriage as between a man and a woman. These bans kept same-sex couples from receiving the substantial government benefits that married couples receive involving things like taxes, adoption and inheritance. In a close 5-4 ruling, the Supreme Court held  marriage is a fundamental right and the 14th amendment requires all states to license same-sex marriages, as well as fully recognize same-sex marriages that lawfully occurred out-of-State.

The Details: 

No union is more profound than marriage, for it embodies the highest ideals of love, fidelity, devotion, sacrifice, and family. In forming a marital union, two people become something greater than once they were. As some of the petitioners in these cases demonstrate, marriage embodies a love that may endure even past death. It would misunderstand these men and women to say they disrespect the idea of marriage. Their plea is that they do respect it, respect it so deeply that they seek to find its fulfillment for themselves. Their hope is not to be condemned to live in loneliness, excluded from one of civilization's oldest institutions. They ask for equal dignity in the eyes of the law. The Constitution grants them that right.--Justice Anthony Kennedy

First, the majority noted that the history of marriage and overall humanity has evolved (i.e. the decline of arranged marriages and coverture, the increase of equality between men and women, the increased acceptance of homosexuality) for the better, "where new dimensions of freedom become apparent to new generations." Over time,  the Supreme Court has recognized the institution of marriage, individual dignity and autonomy as fundamental liberties protected by the Due Process Clause and Equal Protection Clause of the 14th Amendment, which state:

No State shall make or enforce any law which shall abridge the privileges or immunities of citizens of the United States; nor shall any State deprive any person of life, liberty, or property, without due process of law; nor deny to any person within its jurisdiction the equal protection of the laws.

For example, the Supreme Court has invalidated state laws that banned marriage between interracial couples, prohibited prisoners from marrying, prohibited men who were delinquent in child-support payments from marrying, controlled the use of contraceptives and prohibited gay couples from intimate acts.

Extending these principles, the Court noted the inequality of the marriage laws in question: "Same-sex couples are denied benefits afforded opposite-sex couples and are barred from exercising a fundamental right. Especially against a long history of disapproval of their relationships, this denial works a grave and continuing harm, serving to disrespect and subordinate gays and lesbians." With that, the Court held marriage is a fundamental right protected by the 14th Amendment and same-sex couples have the right to enjoy this right as equally as an opposite-sex couple. The Court recognized the option to wait for further legislation, litigation and debate to unfold, but ruled 1. the debate had gone on long enough and 2. "individuals who are harmed need not await legislative action before asserting a fundamental right."

The Bitter, Bitter Dissent(s)

Each of the four dissenting justices wrote their own opinion, Chief Justice Roberts' words alone being longer than the entire majority opinion. Dissenting opinions predictably ranged from arguing the traditional definition of marriage, to the  narrow definition of "liberty" as a fundamental right, to the states' "legitimate" interest of limiting marriages to opposite-sex couples for the purpose of procreation, to explaining that prior cases creating marriage as a right were narrow cases involving opposite-sex couples only, to differentiating between demanding freedom from government action and entitlement to government benefit, to claiming the Constitution gave the majority no authority to rule the way it did.

Roberts went on a weird rant about polygamy, Justice Scalia pretentiously and egotistically criticized the majority for being pretentious and egotistical, while Justice Thomas awkwardly explained that the government didn't take dignity away from people by permitting slavery and internment camps. Several justices characterized the decision as an assault on traditionalists, as if the traditionalists were the victims rather than a gay community that has suffered centuries of humiliation, discrimination, harm, degradation and disrespect: 

I assume that those who cling to old beliefs will be able to whisper their thoughts in the recesses of their homes, but if they repeat those views in public, they will risk being labeled as bigots and treated as such by governments, employers, and schools. --Justice Samuel Alito

All dissenting opinions criticized the majority for what is commonly known as judicial activism, or "legislating from the bench":

When decisions are reached through democratic means, some people will inevitably be disappointed with the results. But those whose views do not prevail at least know that they have had their say, and accordingly are--in the tradition of our political culture--reconciled to the result of a fair and honest debate. In addition, they can gear up to raise the issue later, hoping to persuade enough on the winning side to think again. That is exactly how our system of government is supposed to work." -C.J. Roberts