When two objects begin at rest (at least, relative to each other) or at the same velocity relative to some outside reference (for example Earth), then the speed of both objects, after a given time, will depend on how much acceleration they feel.
F = G M m / d^2
The pull (force) of Earth on the two objects will depend on the mass of the objects. If item A is 5 times the mass of item B, then the force pulling on it will be 5 times more.
HOWEVER, the acceleration depends on the force AND the mass of the object.
F = m a
F/m = a
The force on item A is 5 times more (5 times the gravitational force), BUT the mass is also 5 times more. The two "5" cancel out, leaving the same acceleration for BOTH objects.
On Earth, this would be apparent, except for the air resistance. If you drop a hammer and a feather together, the feather will quickly reach its terminal falling velocity (air resistance = gravity) and take many seconds to reach the ground, while the hammer will rapidly drop on your toe (because you were busy watching the feather).
In space, the major difference is the absence of air resistance.
F = force
G = a constant to make the units work out
M = mass of Earth (in this example)
m = mass of the object being analyzed (item A or item B, separately)
a = acceleration
Because the constant is constant, and so is Earth's mass in this case, the value is often represented as a single thing called GM
F = GM m / d^2
F = m a
On a given object, at a given position, at a given time, both forces are equal. Gravity will cause a force "F" and it is this "F" that causes the acceleration.
m a = GM m / d^2
divide both sides by "m"
a = GM/d^2
The acceleration is the same for all objects because the mass of the small object cancels out.
Newton showed that it does make a difference if the object is very massive -- for example, if you compared a feather with... the Moon. That is because the object is really attracted to the barycentre (centre of mass) of Earth+object. In the case of the feather, the barycentre is still at Earth's centre. However, in the case of the Moon, the barycentre is more than 4000 km away from Earth's centre. This changes the value of "d".