The Supreme Court, 7-2, said the federal law preempted the state law, Adam Liptak explains:
Much of Justice Scalia’s majority opinion concerned the meaning of the phrase “accept and use.” Arizona officials argued that they do accept and use the form, but also require additional information. An airline may accept and use e-tickets, they said, but also require identification....Here's the opinion, Arizona v. Inter Tribal Council. I see that Justice Scalia, writing for the majority, considered using a presumption against preemption that "rests on an assumption... that 'Congress does not exercise lightly' the 'extraordinary power' to 'legislate in areas traditionally regulated by the States.'" (The internal quotes go to Gregory v. Ashcroft, a 1991 opinion that resolved an ambiguity in the federal Age Discrimination in Employment Act to preserve state laws that force judges into retirement at a specified age.)
In the decision on Monday, Justice Scalia said the phrase “accept and use,” when understood in context, meant that the federal form had to be accepted as sufficient.
In the Arizona case, Scalia said that the presumption doesn't apply because Congress enacted the law under the Elections Clause. (Art. I, §4, cl. 1: "The Times, Places and Manner of holding Elections for Senators and Representatives, shall be prescribed in each State by the Legislature thereof; but the Congress may at any time by Law make or alter such Regulations, except as to the places of chusing Senators.”)
Justice Kennedy, concurring, rejected the idea of "a hierarchy of federal powers so that some statutes pre-empting state law must be interpreted by different rules than others, all depending upon which power Congress has exercised." It's still an area of traditional state regulation, and it would make more sense to question the presumption — he says, citing an old Scalia concurrence that did just that — than to minimize the state's concern, given that they are acting in "their own historic role in the conduct of elections." But Kennedy concurs because there's no ambiguity in the National Voter Registration Act that the presumption would resolve. The preemption is clear.
Only 2 justices dissented. Justice Thomas said the Constitution gives the states the exclusive authority over voter qualifications and over whether those qualifications are met, so he adopted the narrower interpretation of the NVRA to avoid an unnecessary constitutional question. Justice Alito said that the Court got it "exactly backwards" when it rejected the presumption against preemption because of the Elections Clause. The Elections Clause manifests a reservation of "default responsibility" to the states, so Congress should have to "speak clearly" to displace the states as they carry out this function the Constitution explicitly assigned to them.
Alito took what had been Sandra Day O'Connor's seat on the court. As Liptak notes, Justice O'Connor was one of the 3 judges who initially heard the case in the 9th Circuit, and she was sitting in the courtroom today as the opinion was announced.