In Chapter 2, Lewis explains how the Supreme Court decides which cases to hear. It's clear and concise and worth summarizing.
Getting cases to the Supreme Court is hard. It doesn't seem like an exaggeration to say the Court works hardest at not deciding cases, and failing that at not deciding them on fundamental constitutional issues.
The first obstacle is jurisdiction. The Constitution limits the Court to hearing only a few types of appeals. The most important of these are cases where an issue of federal law or the Constitution is at stake. These are mostly cases that have been appealed from state courts (in fact, the Court's successful assertion of its right to review state court decisions regarding federal issues was an important early test of federal power).
Even if a federal issue is at stake, a number of precedents and statutes oblige the Court to reject some cases. In the first place, the case must be ripe. That is, federal law requires that all opportunities for appeal must have been exhausted at the state level before the Supreme Court can take a case. In any event, the federal issue must have been raised as early as possible in the case, unless state appellate courts have allowed it in at a later stage. Finally, the court will generally not review cases that were rejected in state courts on procedural grounds.
In addition to these requirements, a litigant must show a direct personal stake in the case. An adverse decision must be shown to have personally caused the litigant some injury. This shows that he or she has standing to bring the case.
All this may not yet be enough. For the Supreme Court is not generally obliged to hear an appeal, even if it meets all of the tests that would permit a review. The Judges' Bill of 1925 granted the Court discretion to choose among the cases it could legitimately take. The Court issues a writ of certiorari when it decides to hear an appeal, and denies the writ, usually without explanation, when it declines a case.
A refusal of certiorari doesn't always reflect satisfaction with the ruling of a lower court. A case with legal merit may be refused because members of the Court don't think, for one reason or another, that it presents the right opportunity to decide the issue. The facts of the case may be too confusing to serve as the basis of a clear decision. Justices may wish to see further exploration and development of the issue in other cases in lower courts before making a ruling. Sometimes a Justice who sympathizes with the claim will vote to refuse certiorari because of a fear that the case will lose, and so create an adverse precedent.