Gym Badges are recognition of how much you've interacted with particular Gyms—through battling, defending, or feeding Pokémon treats.
Badges have four tiers: Basic, Bronze, Silver, and Gold. You’ll unlock new tiers as you interact more with the Gyms. As a reward for your frequent interactions with a Gym, your Badge will earn you bonus items when spinning the Photo Disc.
To view your Badges:
In Map View, tap your Trainer Profile picture.
Scroll down to the Gym Badge section.
Tap the List button to see a full listing of all your Badges.
Tap any Badge to view details about your interactions with the Gym, including your progress towards unlocking a higher tier.
You can also view all your Badges displayed on the globe using the Badge Map button at the bottom left corner of the Gym Badge list.