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Hello,
Air Staff I suppose you are asking about the Flight Crew. Flight Crew are basically pilots. Who are well qualified and trained to fly the plane in any and every condition. The training include weather patterns, flying condition in each country, familiarity with regulations governing safe flight, aviation security, dangerous goods handling, emergency situations etc. in order to conduct a safe flight with all available resources. I'll include the flight dispatchers also in this category who are qualified and trained personnel to prepare every flight and monitor it's safe departure, en route phase and landing. They study the Weather, regulations, notams, MELs etc.to keep it ready before the flight crew arrive for a briefing ahead of each flight.
Management is something related to the back office job where experts sit, plan, discuss, implement and monitor the strategies related to growth and safe operation of the airline.
All the best.