How does the Supreme Court refuse to review a case?

If the U.S. Supreme Court denies a petition for a writ of certiorari (a request to hear a case on appeal), then the decision of the lower court is final.

Denial of certiorari occurs in 98-99% of cases, and in no way implies that the court agrees with the lower court's decision. Denial only means that the case, as presented, isn't of sufficient importance to warrant a review, doesn't involve constitutional issues, conforms to a precedent already set, falls outside the court's jurisdiction, or is moot, etc.

Between 7,500 and 8,500 cases are presented for review each year, but the court can only choose 80-150 to hear, so the Justices have to limit themselves to those cases that have the greatest impact on the law and on society.

If a case is denied certiorari, the decision of the last "court of competent jurisdiction" to handle the case is affirmed and the case is concluded. Affirmation by default does not necessarily indicate the Supreme Court agrees with the lower court decision; nor does accepting a case for review necessarily mean the Court disagrees with the lower court decision.