Intense Sunlight
・Increases the power of Fire-type moves by 50%.
・Allows Solar Beam to be used instantly.
・In Generation V and on, causes Growth to raise Attack and Special Attack two stages each.
・Activates the following Abilities: Chlorophyll, Dry Skin, Flower Gift, Forecast, Leaf Guard, Solar Power.
・Guarantees Harvest will restore a held Berry.
・Changes Weather Ball to a Fire-type move and doubles its power.
・Changes Castform to its Sunny Form and Cherrim to its Sunshine Form.
・Prevents Pokémon from becoming frozen.
・Causes Moonlight, Synthesis, and Morning Sun to recover ⅔ of max HP (twice the normal amount in Generation II).
・Decreases the power of Water-type moves by 50%.
Rain
・Increases the power of Water-type moves by 50%.
・Allows Thunder* and Hurricane to bypass accuracy check.
・In Pokémon Diamond and Pearl, allows Thunder to hit through Protect and Detect 30% of the time.
・Activates the following Abilities: Dry Skin, Forecast, Hydration, Rain Dish, Swift Swim.
・Changes Weather Ball to a Water-type move and doubles its power, and changes Castform to its Rainy form.
・Prevents Pokémon from exploding in Pokémon Mystery Dungeon.
・Decreases the power of Solar Beam and all Fire-type moves by 50%.
・Causes Moonlight, Synthesis, and Morning Sun to recover ¼ of max HP since Generation III, or half the normal amount in Generation II.
Sandstorm
・Activates the following Abilities: Sand Force, Sand Rush, Sand Veil.
・Raises the Special Defense of all Rock-type Pokémon by 50% (Generation IV onwards).
・Changes Weather Ball to a Rock-type move and doubles its power.
・Damages all Pokémon not part Rock-, Steel-, or Ground-type or those who don't have Sand Force, Sand Rush, Sand Veil, Magic Guard or Overcoat as their Ability or Pokémon holding the Safety Goggles.
・Causes Solar Beam to only deal half of its normal damage.
・Causes Moonlight, Synthesis, and Morning Sun to recover ¼ of max HP (half the normal amount in Generation II).
Hail
・Activates the following Abilities: Forecast, Ice Body, and Snow Cloak.
・Changes Weather Ball to a Ice-type move and doubles its power.
・Changes Castform to its Hail Form.
・Allows Blizzard to bypass accuracy check (Generation V onward), or makes its accuracy 100% (in Generation IV).
・In Pokémon Diamond and Pearl, allows Blizzard to hit through Protect and Detect 30% of the time.
・Damages all Pokémon that do not have the Ice-type or the Abilities Ice Body, Snow Cloak, Magic Guard, or Overcoat.
・Causes Solar Beam to deal only half its damage.
・Causes Moonlight, Synthesis, and Morning Sun to recover ¼ of max HP.
Fog
・Doubles the power of Weather Ball.
Note that Weather Ball's type does not change, nor does Castform change form or type
・Lowers the Accuracy of all Pokémon battling by 10%.
・Causes Moonlight, Synthesis, and Morning Sun to only recover ¼ of max HP.
There is also Diamond Dust, although it has no effect in or out of battle.
Source