Supreme Court justices are assigned to each of the circuit courts (also known as the courts of appeals) by the Chief Justice, where they serve as circuit justices. This assignment has a number of duties and powers, but the relevant one here seems to be that the justices can either act on certain requests in a circuit themselves, or refer them to the full Supreme Court.
There are 11 circuits covering all states and most territories, plus two circuits covering Washington D.C. and the federal court system, so justices can be assigned to multiple circuits. The circuits are grouped together based on federal judicial districts and each mainly covers a group of states, though some circuits also cover territories such as Puerto Rico or Guam. All thirteen circuits have an assigned justice, so courts systems in all states and territories have a justice to handle these sorts of decisions.
The Supreme Court's page on circuit assignments, found here, says that Amy Coney Barrett is assigned to the Seventh Circuit, which covers Indiana:
It is ordered that the following allotment be made of The Chief Justice and the Associate Justices of this Court among the circuits, pursuant to Title 28, United States Code, Section 42 and that such allotment be entered of record, effective November 20, 2020.
...
For the Seventh Circuit - Amy Coney Barrett, Associate Justice
(Illinois, Indiana, Wisconsin)
The referenced law covering these assignments is 28 U.S. Code § 42:
The Chief Justice of the United States and the associate justices of the Supreme Court shall from time to time be allotted as circuit justices among the circuits by order of the Supreme Court. The Chief Justice may make such allotments in vacation.
A justice may be assigned to more than one circuit, and two or more justices may be assigned to the same circuit.
I haven't been able to find a good description of all the powers that a justice has when assigned to a circuit, however it seems like they have the ability to decide on emergency requests themselves or refer them to the full court. From an informational article by the Federal Judicial Center, found here:
When Congress created the U.S. circuit courts as the primary trial courts of the federal judiciary, it did not create judgeships assigned exclusively to those courts. Instead, the local U.S. district court judge and two justices of the Supreme Court were to preside in the circuit courts.
...
In 1911 Congress abolished the circuit courts, thereby eliminating circuit riding. The law has continued to require the Chief Justice to allot the justices to the judicial circuits, however, for which they perform occasional duties such as ruling on applications for emergency actions.
And SCOTUSblog, a site dedicated to tracking and reporting the goings on at the Supreme Court, published this article covering the case, prior to the news of Amy Coney Barrett's decision:
The case, Klaassen v. Trustees of Indiana University, goes to Justice Amy Coney Barrett, who is responsible for emergency appeals from the 7th Circuit. Barrett can either act on the request herself or refer it to the full court.
As noted by phoog in the comments, it looks like Rule 22 of the Supreme Court is the relevant rule for what Barrett did here:
- An application addressed to an individual Justice shall be filed with the Clerk, who will transmit it promptly to the Justice concerned if an individual Justice has authority to grant the sought relief.
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- An application shall be addressed to the Justice allotted to the Circuit from which the case arises.
...
- A Justice to whom an application for a stay or for bail is submitted may refer it to the Court for determination.
In short, a request for some sort of emergency stay was submitted to the assigned circuit justice, Barrett, since she had the power to either refer it to the full court or decide on the matter herself (which she did in the case).
