I built one of each aircraft available in my latest game. Only the Aerodrome district is needed to build them.
Re-basing an aircraft can be done to any city, but cities with no Aerodrome can hold just one aircraft. It seems best to build Aerodromes near the front line and especially near unsettled areas that generate Barbarian encampments.
The biplane, fighter and probably jet fighter can be deployed (based) in any hex near its home base. The following turn they can perform an air attack, primarily against land units, since the AI doesn't seem to build any air units.
All air units can perform an air attack from its base (if not deployed nearby), either a city center or Aerodrome district. The bomber and jet bomber can cause signifcant damage against a land or naval unit. The biplane, fighter and jet fighter can also do air attacks against land and possibly naval units too, but do less damage.
The only advanced ranged land unit is the Rocket Artillery unit, but it is a bombard unit and thus has a -17 strength versus land units. The Machine Gun could be used instead, but its ranged strength is only 75 and its range is reduced to 1 from the field cannon's range 2, so it has to get in closer to attack land units. As a result, air units can fill in this gap and they can cover far more territory too.
It is sad that the AI doesn't build (or rarely builds) air units, so there can't be (much) air to air combat, but air units can target either land or naval units, so they do fill a role in the later game, usually a Science Victory or possibly a non-religious Cultural Victory