Have you ever seen a movie or TV show where a local sheriff or detective storms up to a shiny black car with government plates, and almost pops a blood vessel while he yells at the smartly-dressed person who has just begun giving orders (from behind the wheel, no less) to the lowly patrol officers on the scene, only to be rebuffed by said smartly-dressed person?
Well, as an FBI Special Agent, that smartly-dressed person with the federal badge would be you, a member of America's Police.
Aside from having supreme handgun accuracy, Batman-level detective skills, and an arsenal of the coolest gadgets government money can buy, you're also the top law enforcement official wherever you happen to be (source).
You have the jurisdiction to outrank, outflank, and outblank local cops and hometown heavies on cases involving everything from interstate drug trafficking to public corruption, bank fraud to serial killers, and even (especially) on matters of national security.
However, most of that power really only extends to the edge of America's shores. After that it's a mad tussle of jurisdiction between the FBI, CIA, NSA, and other organizations with letters instead of names. At least you'll be able to keep everything alphabetical.