Clarification--I've not gotten past the case summary yet, but Justice Kennedy read the decision based on his opinion, to which Chief Justice Roberts and Justice Alito joined.  Justice Scalia wrote a concurring opinion, to which Justice Thomas joined, and then the Chief Justice wrote his own concurring opinion in response to Sotomayor's pointed critique.  That's the majority in this case.

Here's the Times lead piece:

In a fractured decision that revealed deep divisions over what role the judiciary should play in protecting racial and ethnic minorities, the Supreme Court on Tuesday upheld a Michigan constitutional amendment that bans affirmative action in admissions to the state’s public universities.

The 6-to-2 ruling effectively endorsed similar measures in seven other states. It may also encourage more states to enact measures banning the use of race in admissions or to consider race-neutral alternatives to ensure diversity.

States that forbid affirmative action in higher education, like Florida and California, as well as Michigan, have seen a significant drop in the enrollment of black and Hispanic students in their most selective colleges and universities.

Kennedy's plurality opinion is a narrow one, focusing on the right of the Michigan voters to make policy on affirmative action:

“This case is not about how the debate about racial preferences should be resolved,” he wrote, in an opinion joined by Chief Justice John G. Roberts Jr. and Justice Samuel A. Alito Jr. “It is about who may resolve it. There is no authority in the Constitution of the United States or in this court’s precedents for the judiciary to set aside Michigan laws that commit this policy determination to the voters.”

Sotomayor's Equal Protection point is that the restriction on race based considerations must be compared to other students who get preference for reasons other than objective merit:

[Sotomayor] said the initiative put minorities to a burden not faced by other college applicants. Athletes, children of alumni and students from underrepresented parts of the state, she said, remained free to try to persuade university officials to give their applications special weight. “The one and only policy a Michigan citizen may not seek through this long-established process,” she wrote, “is a race-sensitive admissions policy.” That difference, she said, violates the Constitution’s equal protection clause.

“The Constitution does not protect racial minorities from political defeat,” she wrote. “But neither does it give the majority free rein to erect selective barriers against racial minorities.” Justice Ruth Bader Ginsburg joined the dissent