In DPPt the badges do not have additional effects. I don't remember if the HGSS ones increase stats, but here is what they did in the older games:
KANTO
Boulder Badge (Pewter City) - Attack
Thunder Badge (Vermillion City) - Speed
Soul Badge (Fuschia City) - Defense
Volcano Badge (Cinnabar Island) - Sp. Attack, Sp. Defense
JOHTO
Zephyr Badge (Violet City) - Attack
Plain Badge (Goldenrod City) - Speed
Mineral Badge (Olivine City) - Defense
Glacier Badge (Mahogony Town) - Sp. Attack, Sp. Defense
HOENN
Stone Badge (Rustboro City) - Attack
Dynamo Badge (Mauville City) - Speed
Balance Badge (Petalburg City) - Defense
Mind Badge (Mossdeep City) - Sp. Attack, Sp. Defense
Apart from this, the level obedience and the HMs, there aren't any other effects that I know of.