Writs of Certiorari:
Parties who are not satisfied with the decision of a lower court may petition the U.S. Supreme Court by asking it to grant a writ of certiorari. The Court is not obligated to hear these cases. It only agrees if the case could have national significance, might harmonize conflicting decisions in the federal Circuit courts, or could have precedential value. Four of the nine Justices must vote to accept a case.

Law Clerks:
Each Justice has three or four law clerks per Court term. These recent law school graduates were usually at the top of their class from the best schools. Often, they have served a year or more as a law clerk for a federal judge. Their tasks include legal research to assist Justices in deciding what cases to accept; prepare questions that the Justice may ask during oral arguments; and draft opinions.

Briefs:
If the Justices decide to accept a case, the petitioner and the respondent have to write briefs that state the legal case.

The petitioner and respondent are permitted to file briefs of a shorter length that respond to the other party's position. The Solicitor General can file a brief on behalf of the government. With the permission of the Court, groups that do not have a direct stake in the outcome of the case may also file a brief.

Oral Arguments:
The Court hears oral arguments in cases from October through April, in two-week sessions every month. Oral arguments are heard on Mondays, Tuesdays, and Wednesdays. Oral arguments are open to the public. Typically, two cases are heard each day, beginning at 10 a.m. Each case is allotted an hour for arguments. Lawyers for each party have a half hour to make their best legal case. Most of this time is spent answering the Justices' questions. The Solicitor General usually argues cases in which the U.S. Government is a party.

Conference:
The Justices decide the case at the Justices' Conference, held twice a week. Before then, the Justices frequently discuss the cases with their law clerks.

Only Justices are allowed in the Conference room. First they decide which cases to accept or reject. Then they discuss the cases that were heard since their last Conference. All Justices have an opportunity to state their views on the case and raise any questions. Each Justice speaks without interruption. Each Justice speaks in order of seniority.

When each Justice is finished speaking, they vote. The most senior Justices in the majority and in the dissent write their opinions. If a Justice agrees with the outcome of the case, but not the majority's rationale for it, that Justice may write a second opinion.

Opinions:
All opinions of the Court are handed down by the last day of the Court's term. A majority of Justices must agree to ("sign onto") all of the contents of the Court's opinion before it is publicly delivered.

Source: Supreme Court Procedures
Administrative Office of the U.S. Courts