A writ of certiorari is issued by a higher court when it wants to review a lower court's decision. A writ of mandamus is an order from a court to an inferior government official ordering him or her to properly fulfill his or her official duties. A writ of habeas corpus is a court order to a person or institution holding someone in custody to deliver the imprisoned person to the court. A writ of assistance is a court order instructing a law enforcement official to perform a certain task. A writ of body attachment is a court order directing the U.S. Marshal to bring a person who has been found in civil contempt to the court.

A Writ of Mandamus is a court order directed at a lesser government official requiring that official to carry out the duties and responsibilities of his or her office.

In the United States Court system, the losing party in a court case has the right to appeal to a higher court about the nature and fairness of the verdict. The ability of the a Federal court, particularly the Supreme Court, to review a court decision made by a lower court is called appellate jurisdiction—as in "appeals."

The Rule of Four is a rule that is used in Supreme Court practice to decide which cases to hear. It states that if four or more judges agree that a case should be heard, then it must be heard. This is not a rule codified into law, but rather a precedent established throughout Supreme Court history.

 is the most common length of oral arguments. For more controversial cases, it may be extended. This a procedural arrangement determined by the Justices. They may hear many cases, and thus oral arguments must be concise.

The Supreme Court agrees to hear the case is the correct answer. Remember that “certiorari” is Latin for “to be informed of” which helps us because that is exactly what the Supreme Court is trying to do here. In other words, once the Supreme Court grants cert, they’ll schedule oral argument (over the merits of the case), and then read numerous briefs (essentially written arguments saying why one side should win/lose) in order to get an understanding of the case and thus render a decision.

Nothing cannot be the correct answer because of all of the reasons above. It is likely true that the Supreme Court is interested in the case, but that is only part of the answer.

The petitioner wins/loses cannot be correct because the Supreme Court has only agreed to HEAR the case; they cannot render a decision until after they do so.

The lower court(s) being reversed cannot be true because the Supreme Court may not in fact reverse the lower court: they can affirm (that is, uphold) it. Indeed, oftentimes the Supreme Court will take a case and affirm the reasoning of the lower court in order to clarify a point of law. 

Stare decisis is latin for "let the decision stand" (or thereabouts). It is more than just legal jargon, however, as it is a doctrine on which every common law system rests. More specifically, as a common law system, our courts rely on "precedent" which is a technical term for (similar) cases that have been decided by an earlier court. Reliance on precedent is motivated by more than pure laziness or convenience; it's incredibly important from a stability standpoint.

Take, for example, a case that was over the proper definition of an apple. Imagine that court 1 decided the case in 1800 and held that an apple was "a fleshy fruit, encasing a core with seeds, with a waxy skin that can be any number of colors, but is most commonly red, green, or yellow." Great! Makes sense so far. Imagine further, however, that court 2 heard a case involving the same issue (the definition of the apple) in 1805 and decided that it was "a round pulpy fruit with easily divisible segments, each of which holds seeds." Wait a minute. That sounds more like an orange than an apple, and, more importantly, everything the first court worked for has been turned on its head in 5 years! Now imagine that this scenario repeated itself--litigants and courts would never be certain what the definition of an apple was going to be! Thus, stare decisis, which councils courts to rely on precedent when they can, is of utmost importance to a common law system. 

All of the other answers are incorrect either because the translation is wrong, or because the reason behind the importance is wrong. 

The correct answer is “the process by which the Supreme Court decides if it will hear a case.” Before getting into the meat of the answer, remember that, in regards to the Supreme Courts appellate jurisdiction, it has “discretionary authority.” In other words, the Supreme Court has near-complete control over its docket—it can determine which cases it wants to hear.

Moving on. When a party appeals to the Supreme Court, they petition for a ‘writ of certiorari.’ All of these petitions go into (essentially) a gigantic pile called the ‘cert pool.’ In order to determine which cases will be docketed (and thus heard), the Justices vote. As long as four Justices vote “yes” on a case, the Supreme Court will hear it, hence the “rule of four. “

As for the other answers:

Senatorial courtesy has nothing to do with case selection. Senatorial courtesy is an informal process—a gentleman’s agreement, if you will—in between the President and Congress when selecting district court judges. This also applies to the answer “the process by which the Senate confirms Article III judges/justices.”

“The number of Supreme Court justices . . .  to render a majority decision” is obviously incorrect because there are 9 sitting justices, and 4 is NOT a majority.

“A rule of statutory construction . . . “ is incorrect for a plethora of reasons. For one, statutory construction is not a topic for AP US Gov students. More importantly, it’s completely made up and not true! 

Amicus = “friend,” curiae = “court.” Thus, amicus curiae means “friend of the court.” As for the filing party, it must be a [presumably interested] third party. The two involved parties do not file “amicus” briefs: they file merit briefs (and responses). In order to file an amicus brief, the amici (person or organization seeking to file a brief) must either obtain approval of the parties or of the court. Generally speaking, these interested parties tend to be organizations on one side or the other. In a firearms case, for example, you’ll generally see the Brady Center on one side, and the NRA on the other.

All of the other answers are incorrect for the above reasons.

This is a straightforward question, no tricks or gimmicks. It also requires relatively little explanation, due to its directness, but we’ll go through what each of the barriers is/what it means.

Mootness: This means, essentially, that no matter what the Supreme Court decides, the particular outcome will not change. For example, there was a question of mootness in Roe v. Wade (by the time the case was decided, she already gave birth). Thus, some argued that the case was moot, because no matter what the court decided, Roe could not abort the child (it was already born). The Court reasoned, however, that Roe could become pregnant again, thus the case was not moot. A better example is a case involving racial discrimination and acceptance to, say, law school. By the time the case reaches the Supreme Court, it is likely that the student has been accepted to a different law school, and graduated. Thus his case is likely moot.

Ripeness: This is the opposite of mootness. The Supreme Court will not decide a case that is not yet ripe. In other words, the case has not proceeded far enough along in terms of the litigation stream to allow the court to render a decision.

Standing: This is, arguably, one of the more important barriers to entry. Standing refers to a particularized entity being “injured” (not necessarily physically) by the particular statute at hand. Take, for example, several of the plaintiffs in Heller v. DC. Originally, there were multiple plaintiffs in the case, however, Anthony Heller was the only plaintiff who had applied for a handgun permit, and been denied. The other several plaintiffs attempted to assert that they considered applying, but were too scared/didn’t for whatever reason. The other plaintiffs failed to suffer a redressable injury—they had not been denied of anything! Thus they lacked standing.

Political Question(s): This is probably the most interesting and nebulous one. Essentially, the Court will not render decisions that are better left to the other two branches to decide. A classic example (although now unfortunately no longer applicable) is that of the political division of states into voting units (redistricting). Frankfurter (a Supreme Court justice) wrote a majority opinion in Colgrove v. Green in which the Supreme Court refused to decide a case re: redistricting, saying that the Court “ought not to enter into this political thicket.” 328 U.S. 549, 556 (1946). Frankfurter believed that redistricting was a question better left in the capable hands of the state legislature.