The US Constitution gives the Supreme Court appellate jurisdiction in some cases and original jurisdiction in others. With appellate jurisdiction, the lower court has to hear a case first and then ask the Supreme Court to grant a Writ of Certiorari to hear the case on appeal. However, "[i]n all Cases affecting Ambassadors, other public Ministers and Consuls, and those in which a State shall be Party, the supreme Court shall have original Jurisdiction." These are cases involving cases filed directly with the Supreme Court.

Georgia v. Brailsford is certainly not the only case to take advantage of original jurisdiction. The most famous example might be Marbury v. Madison where Marbury sued Madison to make him deliver his appointment as Justice of the Peace. He filed the lawsuit directly in the Supreme Court based on the authority of the Judiciary Act of 1789. This act was declared unconstitutional by the Supreme Court and Marbury lost since he should have filed it in lower court.

This part is going to get complicated so I apologize in advance. Civil cases under English Common Law were historically separated into two types of courts depending on what you were asking for: Courts of Law and Courts of Equity. Courts of Law generally dealt with issues of money damages (we would think of things like personal injury suits, harm caused by breach of contract, etc.). Equity is asking a Judge to enforce something else - for example, ordering an official to do something (Writ of Mandamus), order someone not to do something (Injunction), declare who has a legal right (Declaratory Judgment), etc. In Marbury v. Madison, for example, Marbury was asking for a Writ of Mandamus against James Madison where the Court would order Madison to deliver the commission he was supposed to do.

I can go into more detail of why this distinction existed, although I'll add the caveat that I'm not an expert in Medieval law and will be referencing my Law School education (so you'll get a better idea of what American Judges thought the history was rather than what the history might actually have been). Suffice it to say that, in the United States, these courts were eventually combined into one court everywhere except from Delaware, which maintains a separate Court of Chancery for matters of Equity. However, the distinction still matters when it comes to juries - you can get juries for actions in law, but not for actions in equity. Marbury v. Madison involved Equity so he didn't get a jury. Georgia v. Brailsford was a lawsuit to recover on a debt owed. Under those circumstances, the questions of fact would typically be decided by a jury while the questions of law would be decided by a Judge, so Chief Justice John Jay impaneled a jury.

Honestly, the strangest thing about the case is that it's the Supreme Court's only jury trial. There are two reasons for this: One the Supreme Court's original jurisdiction is generally concurrent with lower court jurisdiction (i.e., either court can hear it first) and the Court's original jurisdiction is generally discretionary (if you're going to ask for a jury, they're going to say no). After all, the Justices are busy and do not really have time for a jury trial. [ETA: 28 U.S. § 1251 specifies that the Supreme Court has exclusive original jurisdiction for disputes between states and shares original jurisdiction for the other listed categories.] Second, the right to a jury only applies to suits in Common Law while most of the Court's original jurisdiction docket is asking for equitable relief. In a 1950s case regarding the territory of Louisiana compared to that owned by the Federal government (United States v. Louisiana, 339 U.S. 699 (1950)), Louisiana asked for a jury trial on the boundary. The Court denied the request because it was an equity action. Since 1904 with Virginia v. West Virginia, 209 U.S. 514 (1904), the Supreme Court has relied on Special Masters to determine the facts in a case and report those facts to the Justices for approval. Most cases involving original jurisdiction concern the boundaries between states or who has the rights to certain waterways. All these cases are equity cases and do not allow for a jury and the Court usually relies on a Special Master. However, it is worth pointing out that 28 U.S. § 1872 does specifically require questions of fact to be determined by a jury in an original jurisdiction action at law against a citizen of the United States (in other words, if a US citizen is being sued in a case the Supreme Court can hear with their original jurisdiction, facts must be determined by a jury).

Let me know if you have any questions or if you need any part of this explained more clearly. The distinction between law and equity is not an easy concept so I'm happy to go into more detail in an attempt to explain it.

ETA: I added two follow-up posts because this topic interested me quite a bit.