Amicus curiae brief: "Friend of the court" brief; a brief filed by a person, group, or entity that is not a party to the case but nonetheless wishes to provide the court with its perspective on the issue before it. The person or entity is called an "amicus"; the plural is "amici."
Merits brief: Once the Supreme Court has granted certiorari in a case, each party has the opportunity to file merits briefs. Unlike the certiorari-stage briefs, which tell the Court why it should or should not take the case, the merits briefs tell the Court why each party thinks he deserves to win.
Solicitor General: Sometimes called the "tenth justice," the Solicitor General is the lawyer for the U.S. government, and attorneys in that office are responsible for presenting cases on behalf of the United States in the Supreme Court. Someone from the Solicitor General's office will also frequently argue on behalf of the United States when the government is not a party but has filed an amicus brief in the case.
See Glossary of Legal Terms, Scotusblog.