For those not familiar with appellate oral arguments, here's a short primer/FAQ:
Why oral argument? - Appellate oral argument has been described as the Court's "conversation with counsel" about the case and the law. Oral argument can illuminate legal or logical problems not evident from the briefs and which may not otherwise be discovered, distill arguments by testing them, and allow the advocates to respond to the Court’s specific concerns. The U.S. Supreme Court hears oral arguments in nearly all cases it reviews.
What's at issue? - The Supreme Court is a discretionary court, meaning it reviews only cases and issues it want to review. In the course of answering the question presented, the Court may decide related issues. In the ceded lands case, the Court agreed to review the following Question Presented:
In the Joint Resolution to Acknowledge the 100th Anniversary of the January 17, 1893 Overthrow of the Kingdom of Hawaii, Congress acknowledged and apologized for the United States' role in that overthrow. The question here is whether this symbolic resolution strips Hawaii of its sovereign authority to sell, exchange, or transfer 1.2 million acres of state land-29 percent of the total land area of the State and almost all the land owned by the State-unless and until it reaches a political settlement with native Hawaiians about the status of that land.
Questions from the bench - The Court is "hot," meaning the Justices have beforehand read the briefs, applicable law, and precedents, and use the argument to ask questions, usually a lot of them. The advocates generally do not have an opportunity to deliver a scripted argument. This is not like a trial, where the advocates are usually allowed to make an uninterrupted closing argument to the decisionmakers; there is no "testimony" or "evidence." Only the advocates are allowed to speak to the Justices. The Justices' questions are usually very pointed and specific, and may involve what appear to be arcane or technical points. They may also ask hypothetical questions, since the Court is concerned with not only the result in the particular case before it, but the impact of their decision on the law, if any.
Time - The Court generally allocates 1/2 hour per side, for a total of one hour of argument (which may be extended at the Court's discretion, but it rarely is). In the ceded lands case, the federal government has been permitted to present oral argument as amicus ("friend of the court"), meaning that the time split will be maximum 20 minutes for the State, 10 minutes for the United States, and 30 minutes for the Office of Hawaiian Affairs.
How many Justices - Nine Justices sit on the U.S. Supreme Court. Short bios are available here. Chief Justice Roberts is seated in the center of the bench. The Associate Justices are seated in alternating right-to-left of center, according to seniority. Thus, the seating arrangement tomorrow will be (left-to-right): Breyer, Thomas, Kennedy, Stevens, CJ Roberts, Scalia, Souter, Ginsburg, Alito. Justice Ginsburg, who recently underwent treatment for cancer, participated in oral argument on Tuesday, so she will consider the ceded lands case. Pay special attention to questioning from Justice Kennedy - he's been the "swing" vote in many cases recently, and he authored the majority opinion in Rice v. Cayetano, 598 U.S. 425 (2000), the case in which the Court struck down "Hawaiians-only" voting for OHA trustees as a race-based classification by a 7-2 margin.
Decision time - Do not expect a decision from the Court on the day of argument, or any time soon. The Court usually issues its rulings a couple of months after argument.
Petitioner-Respondent-Rebuttal - The arguments usually follow this script: the Petitioner's attorney leads off, generally reserving a few minutes of his or her 30 minutes for rebuttal. After the Petitioner's initial time is up, the Respondent does the same, followed by any rebuttal or reply time reserved. In the ceded lands case, the State will start off, followed most likely by the federal government, then OHA, and then back to the State.
Predictions? - It can be a mistake to base a prediction on the outcome or a Justice's vote based on his or her questions. Often, "devil's advocate" questioning is used to develop the legal and logical contours of an advocate's position, or even to try and communicate via the advocate to a fellow Justice who may not agree with the questioner.