As with any government sector, the US judicial system is ruled by specific nomenclature that distinguishes one type of court from another, as well as other points of style:
The US Supreme Court — US can be spelled out, but there’s no need to do so — should be designated as such, with the initials for “United States,” to distinguish it from state supreme courts even if only the federal court is mentioned. In subsequent references, it can be identified simply as “the Supreme Court” or even “the Court.” (Though court is usually lowercased in generic usage, the word is often capitalized in reference to the highest court in the land.)
Although a state Supreme Court is generally so designated in local media, in publications with more widespread circulation “the California Supreme Court” (or “California’s state Supreme Court”), for example, is preferred. Not all equivalent judicial bodies, however, are so designated; variations include “Court of Appeals,” “District Court,” “Circuit Court,” “Superior Court,” and Court of Common Pleas.”
Regional appellate courts are informally called, for instance, “the Ninth Circuit Court of Appeals,” but it’s better to use the formal title — in this case, “The U.S. Court of Appeals for the Ninth Circuit.” Formal names of district courts follow this form: “U.S. District Court for the Central District of California”; their subunits are divisions whose varying names are capitalized, as in “Eastern Division.”
Names of court cases are italicized, and versus is abbreviated with a v followed by a period: Brown v. Board of Education.
The judicial system is also known by the terms judiciary and, seldom, judicature. The function of the judicial branch of government is to interpret and apply law, as well as ensure equal justice under the law; the legislative system makes laws, and the executive branch enforces them.
The head of the U.S. Supreme Court is designated the chief justice; this job title is capitalized before that person’s name, but a generic identification, even after the person’s name, is “chief justice of the United States.” All other members of the Court are called associate justices; this title is also initial-capped before a name.
A judge is identified by that job title, as in “Judge John Doe,” but remember that when a job title is preceded by a qualifying term, the job title becomes part of a description and is no longer capitalized: “retired judge John Doe,” “appellate court judge Mary Smith.”
And how do you write the form of direct address of a judge? “Thank you, Your Honor,” equivalent to usage for other civil titles — “One more question, Mr. President”; “Please have a seat, Senator.”