While the Gym Leader Badges in the anime are little more than pins with which one can brag, the badges in the video games actually affect the pokemon the badge owner possesses.  Below is a list of the gym badges, the giver of the gym badge, and what that particular badge does to your pokemon.

Monster Balls | Gym Leader Badges | Bonguri no Mi

Kanto Region Badges

Boulder Badge/Grey Badge

The power of all pokemon increases a little.  It lets you use Flash (Flash) any time.  Takeshi gives it to you.

Cascade Badge/Blue Badge

Pokemon up to level 30 obey you.  If they're at a higher level than that, they may not obey your commands.  It also lets you use Iai Giri (Cut) any time.  Kasumi gives it to you.

Thunder Badge/Orange Badge

The speed of all pokemon increases a little.  It also lets you use Sky Jump (Fly) any time.  Machisu gives it to you.

Rainbow Badge/Rainbow Badge

Pokemon up to level 50 obey you.  If they're at a higher level than that, they may not obey your commands.  It also lets you use Super Power (Strength) any time.  Erika gives it to you.

Marsh Badge/Gold Badge

Poken up to level 70 obey you.  If they're at a higher level than that, they may not obey your commands.  Natsume gives it to you.

Soul Badge/Pink Badge

The defense of all pokemon increases a little.  It also lets you use Wave Ride (Surf) any time.  Kyou gives it to you.

Volcano Badge/Crimson Badge

Your pokemon's special abilities increase a little.  Katsura gives it to you.

Earth Badge/Green Badge

All pokemon, no matter what level they're at, will obey you!  Sakaki gives it to you.

Orange Islands Badges (anime-only)

Coral Eye Badge/Sakura Badge

The gym leader from Natsukan Island, Atsushi, gives this badge to Satoshi after he wins a race with his new Lapras.

Sea Ruby Badge/Shiranami Badge

Dan, of the Navel Island Gym, gives Satoshi his second badge after a race down a mountain.

Spike Shell Badge/Rinbou Badge

The Gym Leader of the Yuzu Island Gym, Jigii, gives Satoshi his third Orange Islands badge.

Jade Star Badge/Ruri Badge

Ryuuchin Island's Gym Leader Ruriko gives Satoshi the final Orange Islands badge after an exciting double battle.

Jouto Region Badges

Zephyr Badge/Wing Badge

The power of all pokemon increases a little.  Allows you to use Flash (Flash) any time.  Hayato gives it to you.

Hive Badge/Insect Badge

Pokemon up to level 30 obey you.  If they're at a higher level than that, they may not obey your commands.  It also lets you use Iai Giri (Cut) any time.  Tsukushi gives it to you.

Plain Badge/Regular Badge

The speed of all pokemon increases a little.  It also lets you use Super Power (Strength) any time.  Akane-chan gives it to you.

Fog Badge/Phantom Badge

Pokemon up to level 50 obey you.  If they're at a higher level than that, they may not obey your commands.  It also lets you use Wave Ride (Surf) any time.  Matsuba gives it to you.

Storm Badge/Shock Badge

Pokemon up to level 70 obey you.  If they're at a higher level than that, they may not obey your commands.  It also lets you use Sky Jump (Fly) any time.  Shijima gives it to you.

Mineral Badge/Steel Badge

The defense of all pokemon increases a little.  Mikan gives it to you.

Glacier Badge/Ice Badge

Strengthens your Pokemon's special attack and special defense.  Allows you to use Whirlpool Tide (Whirlpool) any time.  Yanagi gives it to you.

Rising Badge/Rising Badge

All pokemon, no matter what level they're at, will obey you!  Allows you to use Cascade Ascent (Waterfall) any time.

Houen Region Badges

Stone Badge/Stone Badge
The power of all pokemon increases a little.  It lets you use Iai Giri (Cut) any time. Tsutsuji gives it to you.

Knuckle Badge/Knuckle Badge
Pokemon up to level 30 obey you.  If they're at a higher level than that, they may not obey your commands.  It also lets you use Flash (Flash) any time.  Touki gives it to you.

Dynamo Badge/Dynamo Badge
The speed of all pokemon increases a little.  It also lets you use Rock Smash any time. Tessen gives it to you.

Heat Badge/Heat Badge
Pokemon up to level 50 obey you.  If they're at a higher level than that, they may not obey your commands.  It also lets you use Super Power (Strength) any time. Asuna gives it to you.

Balance Badge/Balance Badge
The defense of all pokemon increases a little.  It also lets you use Wave Ride (Surf) any time. Senri gives it to you.

Feather Badge/Feather Badge
Pokemon up to level 70 obey you.  If they're at a higher level than that, they may not obey your commands.  It also lets you use Sky Jump (Fly) any time.  Nagi gives it to you.

Mind Badge/Mind Badge
Strengthens your Pokemon's special attack and special defense.  Allows you to use Diving any time.

Rain Badge/Rain Badge
All pokemon, no matter what level they're at, will obey you!  Allows you to use Cascade Ascent (Waterfall) any time.