In today's announcement from Netherlands Institute for Radio Astronomy, as discussed in a video about two-thirds down this page. At the 1:30 mark, the video says "Gravity itself has mass, which will in turn generate more gravity." Can you please explain this?
Also, at the 0:50 mark of the same video, it states that Einstein said "Gravitational mass is inertial mass". Is this correct, or did it mean to say "gravitational mass is equal to inertial mass."
It is a bit confusing to say that "gravity itself has mass". It would perhaps have been less confusing if they had said, "the source of the gravitational field is energy, and the gravitational field itself has energy. Of course, through the equivalence of mass and energy expressed by $E=mc^2$, it can be said that the gravitational field carries mass-- but there really is no need to say it.
Re your second question, a better way to describe the relationship between inertial and gravitational mass (that is, the mass in $F=ma$ and the mass/energy that's the source of the gravitational field) is that they are the same thing.
Relativity blurs the distinction between mass and energy. This happens even in special relativity - for example it is the reason that a hydrogen atom has a mass smaller than the sum of the electron and proton mass. In effect some of the original mass has turned into energy and been radiated away. This is also the reason for the nuclear mass deficit, and it also means the mass of a gravitationally bound system will be less than the masses of its components.
In general relativity the spacetime curvature is related to an object called the stress-energy tensor, and in this tensor mass and energy can be used interchangeably. They are treated as equivalent and related by Einstein's famous equation $E = mc^2$.
So when we consider the mass of a bound system we need to consider not just the masses of the bits that went into it, but also all forms of energy associated with that system. In my example of a hydrogen atom we have electrostatic potential energy, and in a gravitationally bound system we have gravitational potential energy. However there is also energy associated with the gravitational field itself i.e. energy associated with the curvature of spacetime. We describe this using the stress-energy-momentum pseudotensor.
The point is that the energy of the gravitational field contributes to the total energy of the system and therefore to the total mass of the system. This is what the narrator of your video means when they say gravity itself has mass.
Now on to your second paragraph.
The equivalence principle states that gravitational and inertial mass are equivalent. You ask whether the two are actually the same thing, or whether they just have the same value, but this isn't really a meaningful question as the mass is just a parameter and the theory doesn't care whether they are same or just have the same value - the end result is the same.
The first experiment to rigorously test the equivalence principle was the Eötvös experiment, which showed that different materials behaved the same way in a gravitational field. This is more subtle that it seems at first sight because different elements have different electrostatic and nuclear binding energies. So the experiment is not just showing that mass due to the rest masses of electron, protons and neutrons behaves the same way, but also the contribution of the mass from the electrostatic and nuclear binding energies behaves the same way.
In the experiment described in the video there is also a contribution to the mass from the energy of the gravitational field. What's more this contribution is different for the neutron star and white dwarf because spacetime is more highly curved around the neutron star. So this experiment is showing that the contribution of the mass due to the energy of the gravitational field also obeys the equivalence principle. It is taking the Eötvös experiment and extending it to include the energy of the gravitational field as well.
The reason this is so interesting is because there are a number of theories that extend general relativity. GR is in a sense the simplest theory that explains gravity, but we can construct more complicated theories of gravity such as Brans-Dicke theory. There are lots of such theories and in some of them the equivalence principle is violated for the energy of the gravitational field. This experiment sets a lower limit on this violation and therefore rules out some of these competing theories.
