The value of the gravitational acceleration on the surface of a planet of radius R and mass M is g = GM/R². So the gravity on the moon is 1/6th that of Earth because the moon is far less massive than the Earth and has a different radius(R) as well. The mass of the moon is 1/82nd of the earth's and the radius of the earth is 6300 km whereas for the moon its only 1738 km.

There are motions in the atmosphere and oceans because of the rotation of the earth and because of the gravity. If there was "no Gravity" the earth would not be here! The earth accreted from a bunch (trillions upon trillions) of rocks. What forced these rocks to accrete? GRAVITY!!!!!!!!!!

Every object in the Universe, planets, stars etc., and also smaller objects such as moons exert a gravitational pull. This means that other objects will be drawn towards them. The strength of the attraction is directly related to the size of the planet or moon. Therefore as the moon is a fraction of the size of the earth it's gravitational pull is 1/6th as strong as that of the earth. Gravity is a basic property of every object in the universe and cannot be 'shared' between objects as your question suggests. We expect the moon to have less gravity because it is smaller, the force felt by people walking on the moon is independent to any force felt by people walking on the earth.

If there was no gravity on earth there would be no atmosphere (i.e. no air to breath) so there would not be wind at all. There would also be no life on earth!!

Weather patterns such as hurricanes and tornadoes are caused by complex interactions between rising and sinking air (convection) as a result of the sun's heat. You could look into this in more detail on the web probably. They are not affected by gravity however, gravity is a constant force.

The gravitational acceleration that we feel on the surface of the earth is (approximately) governed by the following equation:
a = GM/R² where:
G = the Gravitational constant (which presumably is the same everywhere in the universe)
M = mass of the Earth (in kilograms, for example)
R = radius of the Earth (in meters, for example).

For the Earth, if you plug in the appropriate numbers, you get acceleration equals 9.8 meters per second squared. Since the moon or another planet or the sun have different values for mass and radius, you would get a different answer for the gravitational acceleration on the surface of that object.

For example, the Moon's radius is 0.27 of the earth's and the moon's mass is 0.012 of the earth's mass. So if you take 0.012/(0.27)² you get 0.16 which is about 1/6.

If there was no gravity on earth there wouldn't be hurricanes or tornadoes because there wouldn't be an atmosphere! Gravity is necessary to keep the atmosphere from literally floating away.

The acceleration of gravity on the surface of the earth (or any spherical planet) depends both on the planet's total mass and its density.

To a good approximation the acceleration
a = G*M/(R*R) where G is the gravitation constant, M is the mass and R is the radius.
So, on the moon, the mass is much less, but the radius is also much lower, giving the moon a surface acceleration of about 1/6 that of the earth.

Since the two bodies are so close (238,000 miles)lunar gravity does affect the earth, mainly this is visible in the tides. (Earth's gravity keeps the moon in orbit around the Earth). Without gravity, there could not be hurricanes and tornadoes since both rely on energy gained from falling air masses.