What causes objects to fall on earth and what is the rate at which this occurs?

Newton called it "Universal Gravitation". 
Now, most people call it "gravity".

What gravity does is: 

It causes there to be forces that pull every speck of mass in the
universe toward every other speck of mass in the universe.

-- If you just think about two gobs of mass, then there are two forces
of gravity. 

-- One force pulls gob-A toward gob-B, and the other force
pulls 'B' toward 'A'. 

-- The strengths of both forces are EQUAL. 

-- The strength depends on the product of the two masses. 

It doesn't matter if the masses are the same or if one is ginormous
and the other is minuscule. It's the product that counts. 
Bigger product ==> stronger force.
Smaller product ==> weaker force.

-- The strength also depends on the distance between the two masses. 
Farther apart ==> weaker force. 
Closer together ==> stronger force.

-- When you hold a rock in your hand, there are forces of gravity
between the rock and the Earth.

-- When you let go of the rock, one force pulls the rock toward
the Earth.    You call that force "The weight of the rock".

-- The other force force pulls the Earth toward the rock.

-- The forces are equal.   The rock's weight on Earth is equal to
the Earth's weight on the rock.

-- Force applied to a mass causes the mass to accelerate in the
direction of the force.

-- The force on the rock causes it to accelerate toward the Earth.
We call that the act of "falling".

-- The force on the Earth causes it to accelerate toward the
rock.  Nobody ever notices that, nobody ever talks about it,
and you are thinking that I am full of merren tzimmes.  But it's
true. I swear it.  You could look it up.  Furthermore, I laugh at
your derogatory remark, because it is often true and my wife's
merren tzimmes is delicious.

-- The reason we never notice it is:  When equal forces are applied
to two different masses, the smaller mass accelerates more, and the
bigger mass accelerates less. 

When you allow the 1-kilogram rock and the Earth to accelerate
toward each other under the influence of the equal gravitational
forces between them, they do indeed accelerate toward each other. 

BUT ... the mass of the Earth is roughly  

               5,972,000,000,000,000,000,000,000 times

the mass of the rock, so the acceleration of the rock is roughly

               5,972,000,000,000,000,000,000,000 times

the acceleration of the Earth.

So the falling of the rock toward the Earth is substantially more
obvious than the falling of the Earth toward the rock.

-- The rate of acceleration of any object falling on or near
the Earth's surface is 9.8 m/s² .

That means that the object's speed increases steadily and
continuously.  At any instant, the speed at which it's falling
is 9.8 meter/sec faster than it was 1 second earlier.