Scalia/Thomas: The two clauses of the Second Amendment (the "militia clause" and the "right to bear arms clause") are not contradictory. As Scalia said in argument today, the entire Amendment can essentially be read as "because we think a militia is so important, people should have the right to bear arms." Under this reading, Scalia and Thomas would define the right to bear arms as a fundamental right and require strict scrutiny.
Roberts: The two clauses are severable, and there is therefore a private right to bear arms. Roberts never specifically articulated the right as a "fundamental" one, nor did he necessarily describe the level of scrutiny to be applied. He did, however, throw out several different restrictions he believed would be perfectly reasonable: restricting access to machine guns, to plastic guns that could pass metal detectors, or restricting handgun ownership to person over the age of 21.
Kennedy: The two clauses are severable, and there is therefore a right to private ownership of firearms. Specifically, the Amendment was crafted to ensure a right of self-defense. The Amendment was drafted as a means of allowing settlers to protect themselves from animal and Indian attacks. Kennedy also appeared to suggest some sort of intermediate scrutiny.
Breyer: There is a right to private handgun ownership, but the militia clause should be read to mean gun ownership is subject to governmental regulation. Breyer seems to suggest a Rational Basis review, and would find the D.C. law to be reasonable.
Stevens/Souter: The militia clause and the right to bear arms are not severable. Any private right to bear arms is limited to the context of a state-regulated military.
Given the way things played out today, it seems apparent that the majority opinion will be authored by either Roberts or Kennedy (I got the impression it would be Kennedy when reading transcripts), with a concurrence by Scalia arguing for Strict Scrutiny, a concurrence/dissent by Breyer, and a dissent by Stevens. Mike O'Shea over at Concurring Opinions came to essentially the same conclusion, and explained the breakdown in greater detail. SCOTUSBlog has quite a bit of coverage and links to numerous opinion articles. And of course The New York Times weighed in, but didn't really say much besides a generic summary. Dave (friend of this blog) also, as usual, has quite a bit of insight to lend. I am not sure if he plans to update his Traditional Notions with his take on this case, but readers are encouraged to check in the coming days, or to simply talk to him around campus.