With an FAA Private Pilot Certificate, you are allowed to fly small, single-engine aircraft for personal and recreational purposes. These typically include light aircraft like Cessna 172s or Piper Cherokees. You are not allowed to fly aircraft that require a more advanced certification or for commercial purposes unless you have additional ratings. If you wish to fly more complex or multi-engine aircraft, you will need further training and ratings, such as the Multi-Engine Rating or Complex Aircraft Endorsement.