Short answers, pulled from the story.
The Supreme Court of the United States is the highest court in the federal judiciary. It has ultimate appellate jurisdiction over all U.S. federal court cases and over state court cases that turn on questions of constitutional or federal law. It also has the power to strike down acts of Congress or presidential directives that violate the Constitution.
The Supreme Court has nine members: one chief justice and eight associate justices. The president nominates justices with the advice and consent of the Senate. Justices hold lifetime tenure, serving until they die, retire, resign, or are impeached and removed.
The Supreme Court held its inaugural session from February 2 through the 10th of February 1790, at the Royal Exchange in New York City, which was then the U.S. capital. The court's first cases did not arrive until 1791, so those early sessions were devoted to organizational proceedings.
Marbury v. Madison was decided in 1803 under Chief Justice John Marshall and established the principle of judicial review, giving the Supreme Court the power to invalidate acts of Congress that violate the Constitution. It remains one of the most consequential rulings in the court's history.
The court receives roughly 7,000 petitions for writs of certiorari each year and grants about 80. At least four of the nine justices must vote to accept a case. The court's Rule 10 directs it to look for cases that resolve conflicts between federal circuit courts, correct major departures from accepted procedure, or address important unresolved questions of federal law.
Only one justice in the court's history has been impeached: Samuel Chase, in 1804. The House of Representatives adopted eight articles of impeachment against him, but the Senate acquitted him. Chase remained on the court until his death in 1811.