A brief is a written legal document that is presented to the court arguing why a party to the case should have a decision in their favor.
Headnotes are summaries of the key legal points. Headnotes are useful for a quick understanding of the decision, but they are the editor's remarks and not the court's. Footnotes, however, are written by the justices on the court.
An Opinion is an explanation of the decision of the court. In the Supreme Court opinions are catagorized thus:
A syllabus summarizes the points decided in the case.
If you want to look up more in depth background information about Supreme Court cases, here are a few online books:
Find briefs, dockets, joint appendixes, oral arguments, opinions and other documents for U.S. Supreme Court cases from 1975-present. A good place to search to find all briefs, summaries, opinions, etc. organized by case on one screen. Limit to Landmark Cases, and you can search for cases by subject including "Right to Privacy," "Fourth Amendment," "Due Process of Law," and the like. INCLUDES BRIEFS.
Academia-created Supreme Court website. Search for cases by name, or by court term, and status before the court. Includes Oral Argument audio when available.