Gubernatorial means relating to a governor or the office of governor. When news reports discuss a gubernatorial race, they're talking about an election where candidates compete to become their state's governor. A gubernatorial candidate is someone running for governor, and gubernatorial duties are the responsibilities that come with the job.

You might see headlines about “gubernatorial debates” before an election, or hear about a governor's “gubernatorial term” (the period they serve in office). While we could just say “governor's race” or “governor debate,” gubernatorial is the proper adjective form, just like presidential relates to president and congressional relates to Congress.

The word sounds formal and official because it is: it appears in news coverage, political discussions, and legal documents. When you hear gubernatorial, you know the topic involves serious state-level politics and leadership.