Yes FBI agents do have powers of arrests and can arrest you for violation of some law in their presence or if you violate some federal law.
Most times local law enforcement will be the ones to arrest you even though the FBI has powers to arrest you but most times FBI will only make arrests if they want you in Federal Custody and not just local police custody.
Usually you'll have to commit a Federal Offense before the FBI will make an arrest but they can arrest you if they need too.