The case that Obama wants to overturn is Michigan v Jackson and you can read it here.
From the AP
The Justice Department, in a brief signed by Solicitor General Elena Kagan, said the 1986 decision "serves no real purpose" and offers only "meager benefits." The government said defendants who don't wish to talk to police don't have to and that officers must respect that decision. But it said there is no reason a defendant who wants to should not be able to respond to officers' questions.
At the same time, the administration acknowledges that the decision "only occasionally prevents federal prosecutors from obtaining appropriate convictions."
The administration's legal move is a reminder that Obama, who has moved from campaigning to governing, now speaks for federal prosecutors.
The administration's position assumes a level playing field, with equally savvy police and criminal suspects, lawyers on the other side of the case said. But the protection offered by the court in Stevens' 1986 opinion is especially important for vulnerable defendants, including the mentally and developmentally disabled, addicts, juveniles and the poor, the lawyers said.
"Your right to assistance of counsel can be undermined if somebody on the other side who is much more sophisticated than you are comes and talks to you and asks for information," said Sidney Rosdeitcher, a New York lawyer who advises the Brennan Center for Justice at New York University.
Stephen B. Bright, a lawyer who works with poor defendants at the Southern Center for Human Rights in Atlanta, said the administration's position "is disappointing, no question."
Bright said that poor defendants' constitutional right to a lawyer, spelled out by the high court in 1965, has been neglected in recent years. "I would hope that this administration would be doing things to shore up the right to counsel for poor people accused of crimes," said Bright, whose group joined with the Brennan Center and other rights organizations in a court filing opposing the administration's position.
Former Deputy Attorney General Larry Thompson and former FBI Director William Sessions are among 19 one-time judges and prosecutors urging the court to leave the decision in place because it has been incorporated into routine police practice and establishes a rule on interrogations that is easy to follow.
SCOTUSblog has more, as does Volokh
Well, Obama and company seem to be consistant. They hate all of our freedoms.