A Gym Badge is the award for defeating a Pokémon Gym Leader, traditionally one of eight in a region, and badges historically raised traded Pokémon's obedience level, unlocked field moves, and in some games boosted a stat. Collecting all eight usually grants access to the Pokémon League.
In the classic structure from Generation I through most later games, a region has eight Gyms, each built around one type and led by a Gym Leader with a small puzzle or gimmick between you and the battle. Beat the Leader and you receive the badge, a TM in most generations, and permission tied to that badge. Gym order was mostly fixed in early games; Pokémon Scarlet and Violet let you challenge its Gyms in any order, though their teams do not scale with you.
The obedience function is the badge's real teeth. A traded Pokémon above the level your badges cover may ignore orders, use a different move, or fall asleep mid battle. In Generation I badges also quietly boosted one stat each, and badges long controlled which Hidden Machine moves you could use outside battle, such as Surf or Fly, until field moves were retired after Generation VI.
Not every game uses the structure. Pokémon Sun and Moon replaced Gyms with Island Trials and captains, and Pokémon Sword and Shield turned Gym battles into stadium events with Dynamax. The badge count still works as the game's progress bar either way: players describe where they are in a run by how many badges they hold.
In play"With four Gym Badges my traded Charmeleon will finally listen to me up to level 50."