In less technical language, what exactly is a "gravitational form factor"? The term "gravitational form factor" is a term I don't recall ever seeing before the year 2018 (about three decades after I started reading physics papers).
I have read several recent papers about them, most recently, this one, claiming to have experimentally determined a gravitational form factor for a pion, and also this one predicting gravitational form factors for spin-0 hadrons, and this one, touching upon energy momentum form factors. This paper from 2016 (which I looked up today) is a bit more clear, but still leaves me less than 100% sure that I have the concept down.
I had thought I knew what a form factor meant outside of the gravitational context in particle physics (see also this previous answer at Physics.SE), but after reading those papers, I'm not sure I really understood that either. After reading these papers, I'm still fuzzy on just what a gravitational form factor is, or means, or in what kind of application you would use it.
Could someone explain the concept in less technical language than is used in these papers what a "gravitational form factor" is, what it means, and how it would be used?
 A: A qualitative answer by an experimentalist:
This article gives the description in words of why form factors are used:

How BIG ARE the elementary particles? How is their charge distributed? These questions are tackled with form factors, which are measures of the charge and magnetic‐moment distributions in the particles. Scattering of electrons on nucleons, and recent measurements made with electron–positron colliding beams, give form factors for the proton, neutron and pion. 

Elementary particle experiments are scattering experiments that try to define the interactions of tiny composite entities obeying quantum mechanical and special relativity algebras. These are studied in the energy momentum space, and the measurements can, using Fourier transforms, give information about space, and thus give a measure of the size of the composite particles.
In this answer about form factors , the format and use of form factors to decide on the size of composite objects is described. 

In more mathematical terms, the cross section for this scattering is given by the Rosenbluth formula $$\sigma =\sigma _{0}\left[ W_{2}+2W_{1}\tan ^{2}(\frac{
\theta }{2})\right]$$ where $\sigma_0$ is the classical cross section (Rutherford for spinless particles, Mott for spin-1/2 particles) and $W_1$ and $W_2$ are the form factors. A particle is called point-like if the form factors don't depend on the momentum transfer $Q^2$. Otherwise, the size of the particle is related to the Fourier transform of the form factors.

It seems that this gravitational form factors business is extending the notion by trying to see the effect of gravitational interactions on the form factors, and because of general relativity,this means  space distortions are adapted to   the form factor tools.
You ask:

what a "gravitational form factor" is, what it means,

In analogy it should give the form of the mass distribution of a complex, quantum mechanical system, as probed by the gravitational interaction. 

and how it would be used?

in validating a general relativity model , and giving a new tool for studying hadron components.
The strong force , due to its high couplings cannot be used to probe quarks within the hadron, the way one uses charge distributions to probe the behavior of hadrons within a nucleus. Due to the high complexity of a hadron, see an illustration for a proton here, the electromagnetic form factor is highly complicated. The gravitational interaction gives a  tool to measure the form of a hadron in analogy.
