Final answer:
An amicus curiae brief is what interest groups submit to influence Supreme Court decisions. These briefs provide insights and legal arguments from stakeholders not directly involved in the case but affected by its outcome.
Step-by-step explanation:
The type of brief that an interest group might submit to influence the Supreme Court's decision is a amicus curiae brief. This term is Latin for "friend of the court."
An amicus curiae brief is often submitted by someone who is not a party to the case but has a strong interest in the matter's outcome. These briefs offer insights, legal arguments, or expertise that might help the court make a more informed decision.
Interest groups use amicus curiae briefs as a way to lobby the judiciary. Unlike lobbying in legislative or executive branches, they can't affect the writing or passage of laws directly in the courts; hence, they submit these legal documents to present their viewpoints on cases that the courts are hearing, which can potentially sway the decisions of the judges and justices.
A notable case that saw an influx of amicus curiae briefs is Obergefell v. Hodges, which resulted in the legalization of same-sex marriage across the United States.