An Air Battle is a type of a Pokémon battle featured only in the anime. Debuting in Cilan Takes Flight!, Skyla invented and followed this battle mechanic in her Gym Battles, until the events of the mentioned episode.
Prior to the events of Cilan Takes Flight!, Skyla battled in the traditional manner and was quite good. As a result, she received floods of challengers each day. However, this left her with little time to pilot her plane and do her air shows. So she invented Air Battles as a quick method of battling, wherein, Skyla determines in her mind, whether her Pokémon would be able to overwhelm the challengers', based on type advantages and disadvantages and the combination of moves. If she felt her team would have a disadvantage, she would give the challenger the Jet Badge. However, if she felt her team had the advantage, she would deny the challenger a Badge. This method worked well and allowed her time for her flight hobbies, until Ash and Cilan questioned the credibility and effectiveness of the method. As a Gym Leader himself, the Air Battles were simply unacceptable to Cilan and he challenged Skyla to Gym battle.
Skyla decided to battle Cilan first seeing that it would be the first time she battled another Gym Leader. She performed her "Air Battle" first and concluded that she was the winner. However, Cilan persuaded her to battle the traditional way. Like she had predicted in her head, she came out victorious in the real battle and thus did not feel she needed to change her methods at all.
In order to help make her realize the true meaning of being a Gym Leader, Ash battled Skyla himself in An Amazing Aerial Battle!. He eventually won with his Tranquill evolving into Unfezant, earning himself the Jet Badge. She then learned not to take Gym battles so lightly and promised to be a proper Gym Leader from then on, thus disbanding Air Battles.