Well, I'm not sure where you got those names but the descriptions are half right.
Rain Dance:
Causes rain to start falling.
Boosts Water type moves, weakens Fire type moves, and Thunder has 100% accuracy.
Automactically starts when Kyogre comes into battle because of the ability Drizzle.
Pokemon with the ability Swift Swim have doubled Speed.
Pokemon with the ability Rain Dish regain HP every move that it rains.
Sunny Day:
Causes the sun to shine brightly.
Boosts Fire type moves, weakens Water type moves, Solarbeam charges and releases in one turn.
Automatically starts when Groudon eneters battle because of the ability Drought.
Pokemon with the ability Chlorophyll have doubled Speed.
That's all I know.