Obviously, the trail to the Supreme Court of the United States is slightly different for each case that makes it there, but SCOTUS has provided a theoretical example on its blog to illustrate how the procedure works. The example begins with a man suing his employer, claiming that his rights under the Constitution and the federal Civil Rights Act were violated.
The lawsuit would begin in a federal district court. These are the lower courts responsible for hearing issues and conducting trials on federal matters. If the man were not happy with the decision made by the federal district court, he could appeal the case. The appeal would be reviewed by one of 13 appellate courts in the country.
During the appellate review, a panel made up of three federal judges reviews the case. Often, this is conducted in writing only. The judges make decisions on matters of law regarding the case. Depending on the outcome of the appeal, the man can either file for a rehearing or petition the Supreme Court to review the case.
If the man decides to petition the Supreme Court, he has 90 days from the appellate court decision to file what is called a petition for certiorari. A motion can be filed requesting more time to file the petition if a need exists. The opposing party in the case can forego the right to file anything regarding the petition, can agree to the petition or can file a brief to oppose the petition.
After all briefs related to this process are filed, the Supreme Court makes a decision about hearing the case. Only then might the high court actually review the case. While most cases don't go all the way to the Supreme Court, working with a lawyer who understands appeals processes is important if you are seeking to overturn a lower decision.
Source: SCOTUS Blog, "Supreme Court Procedure," accessed Jan. 15, 2016