What is meant in mass' definition 'resistance to acceleration'? Wikipedia:

Mass is both a property of a physical body and a measure of its
  resistance to acceleration

What does this mean?  I know of the $E=mc^2$ formula.  I think, the energy required to 'accelerate' an object is the energy required to change said object's gravitational field.
More on this: This acceleration creates a 'kink' in the gravitational (or another eg. Electromagnetic) field, and I believe the energy of this 'kink' is that of the change in energy of the object.
This train of thought explains why a photon has momentum but no mass, as it's only an EM wave, not spanning to infinity, therefore changing its direction (accelerating it) does not require creating a 'kink' in it's EM wave, and therefore has no mass.  This assumes that a photon emits no gravitational field.
Is this train of thought correct?
If not, what is this 'resistance to acceleration'?
 A: 
I think, the energy required to 'accelerate' an object is the energy required to change said object's gravitational field.

As jd_pm says, there are two physics definitions involved in what is called "mass". The inertial mass of classical newtonian gravitation, 
$F=ma$  where m is the acceleration
and the invariant mass of special relativity  which is the "length" of the four vector, and is invariant to Lorenz transformations:

The invariant mass (or rest mass) is the one in use in particle  physics, because it identifies particles and composite systems uniquely.
The m in $E=mc^2$ is a third m, that is no longer used as it introduces the confusions seen in your post. It is the relativistic mass and is not a constant, it is a function of the velocity:

It includes both the kinetic energy and rest mass energy for a particle. The kinetic energy of a high speed particle can be calculated from


The link has the algebra that connects the two concepts.
The relativistic mass, is the instantaneous inertial mass of an object and is useful only in case of calculations for high speed objects, like rockets and speceships, where the inertial mass is necessary to see how much fuel is necessary to accelerate further.

I think, the energy required to 'accelerate' an object is the energy required to change said object's gravitational field.

Here is another confusion, because you are now using the concepts of General Relativity.
Physics is not about beliefs and hand waving speculations, but about coherent mathematics. Better sit down and study the mathematics of General Relativity and the differences in the concept of gravity introduced by general relativity than to speculate.  
A: Well from a classical Newtonian perspective, F=Ma, it's pretty clear. If you apply the same force to two different objects of different masses, the one with less mass will have a greater acceleration.
Energy wise, we know from $E=\frac{mv^2}{2}$ that if you give the same amount of energy to two different objects at rest, the one with the less mass will acquire the greatest speed. 
A: You are asking for information about the inertial mass. It is the property of an object that determines its resistance to change its velocity (i.e. to be accelerated).
It is important to distinguish between inertial and gravitational mass.
The gravitational mass determines the strength of the gravitational attraction between two objects.
