What is the relationship between gravity and inertia? What is the relationship between Gravity and Inertia?  Einstein told us that gravity and inertia are identical.  And from the fact that two different masses fall at the same rate, I believe we can say that gravity and inertia are equal (That is, the inertia of a dropped larger mass is exactly sufficient to slow it’s acceleration to the same level as a dropped smaller mass, regardless of them being dropped on the Earth or on the Moon).   But is this where we are left hanging: that gravity and inertia are both identical and equal?  Is gravity inertia?  Or is inertia gravity?  What is the next step beyond saying that gravity and inertia are both identical and equal?
 A: 
Einstein told us that gravity and inertia are identical.

Yes, Einstein did say that gravity and inertia are identical, despite people in the comments telling you to the contrary.  This is a common error derived partly by Einstein’s equating of gravitational mass with inertial mass (in his principle of equivalence), but mostly simply because gravity and acceleration look like different phenomenon.
You could say that gravity and inertia are identical, and that the gravitational field and acceleration are inductive pairs (similar to the electromagnetic field and electric current.)  A gravitational field induces acceleration, and acceleration induces a gravitational field. 
From Einstein’s 1918 paper: On the Foundations of the General Theory of Relativity…
 http://einsteinpapers.press.princeton.edu/vol7-trans/49
“Inertia and gravity are phenomena identical in nature.” - Albert Einstein 
In a letter Einstein wrote in reply to Reichenbacher...
.http://einsteinpapers.press.princeton.edu/vol7-trans/220
“I now turn to the objections against the relativistic theory of the gravitational field. Here, Herr Reichenbacher first of all forgets the decisive argument, namely, that the numerical equality of inertial and gravitational mass must be traced to an equality of essence. It is well known that the principle of equivalence accomplishes just that. He (like Herr Kottler) raises the objection against the principle of equivalence that gravitational fields for finite space-time domains in general cannot be transformed away. He fails to see that this is of no importance whatsoever. What is important is only that one is justified at any instant and at will (depending upon the choice of a system of reference) to explain the mechanical behavior of a material point either by gravitation or by inertia. More is not needed; to achieve the essential equivalence of inertia and gravitation it is not necessary that the mechanical behavior of two or more masses must be explainable as a mere effect of inertia by the same choice of coordinates. After all, nobody denies, for example, that the theory of special relativity does justice to the nature of uniform motion, even though it cannot transform all acceleration-free bodies together to a state of rest by one and the same choice of coordinates.” - Albert Einstein 
From Albert Einstein’s book: The Meaning of Relativity, pg 58 
“…In fact, through this conception we arrive at the unity of the nature of inertia and gravitation. For according to our way of looking at it, the same masses may appear to be either under the action of inertia alone (with respect to K) or under the combined action of inertia and gravitation (with respect to K’). The possibility of explaining the numerical equality of inertia and gravitation by the unity of their nature gives to the general theory of relativity, according to my conviction, such a superiority over the conceptions of classical mechanics, that all the difficulties encountered must be considered as small in comparison with the progress.” - Albert Einstein 
Here and in other places Einstein specifically emphasizes the equivalence of gravity and inertia, and not merely the equivalence of gravitational and inertial mass.

… But is this where we are left hanging: that gravity and inertia are
  both identical and equal? Is gravity inertia? Or is inertia gravity?

Yes, that is kind of where we are left hanging.

What is the next step beyond saying that gravity and inertia are both
  identical and equal?

The next step would be in solving in greater detail the physics of inertia.  You can search for things like “source of inertia” to get an idea of how some physicists in the past have approached this problem.  My feeling is that when the mystery of inertia is more or less solved, Einstein’s assertion on the equivalence of gravity and inertia will be validated. 
A: 
Einstein told us that gravity and inertia are identical. And from the fact that two different masses fall at the same rate, I believe we can say that gravity and inertia are equal ...

As in the comments, this is not an accurate rendering of the equivalence principle as Einstein first pondered it. Rather, that "gravitation and inertial mass are equal" is how it should read. There are two distinct characteristics of a body - gravitational mass and inertial mass. The former measure's a body's "coupling strength" to a gravitational field as Newton conceived it - it measures the how much force a "standardized" gravitational field exerts on a body. The latter measures a body's "resistance to shove"; it measures how much impulse you need to impart to a body to change its velocity by a standardized amount. In more experimental terms: the former measures how much a body will stretch a spring balance when hung from the balance in a standardized gravitational field. The latter is to do with how quickly a body moves after it is shoved by a given standardized impulse shoving machine. On the face of it, these are very different experiments and two very different properties. And yet, bodies of different inertias fall at the same acceleration in a gravitational field. If this really is true, then the only way that this can happen is if the two different properties - inertial mass and gravitational mass - are precisely proportional to one another. We can then arrange our definitions so that the proportionality constant is unity and call the two equal. But the key result that allows this equality is proportionality, and the demonstration of proportionality was the result confirmed by the Eötvös experiment.

What is the next step beyond saying that gravity and inertia are both identical and equal?

After a great deal more pondering, this lead Einstein to the general theory of relativity. In many lay explanations it is often implied that the principle of equivalence is the key result that lead to GTR and that GTR should somehow leap out as obvious to the reader from it. This is not at all true. Equivalence was a very early hint. Having been the trumpeted all-important subject in Einsteins early papers of about 1907, it shrinks back into the background thereafter and its presence in GTR is actually quite subtle. 
One way of dealing with the hint of equivalence is to reflect that there is another important situation in classical physics where force on a body is proportional to its mass and that is in non inertial frames of reference (such as in the often touted constantly accelerating space elevator). From the standpoint of a non-inertial observer, bodies experience forces of no obvious source in proportion to their inertial mass, exactly as happens for gravity. 
So perhaps the surface of the Earth is not an inertial frame? Indeed in classical general relativity, this is exactly what is happenning. General relativity postulates that space and time form an in general curved (in a very technical sense - don't expect to grasp this notion with simple visual pictures; see also here) manifold and that the motion of free bodies is along geodesics in this manifold. If something is not moving along a geodesic, then a force in proportion to its inertial mass must act on to give rise to this non-geodetic motion. Moreover, general relativity postulates that a generalized notion of energy gives rise to this curvature. So, at the surface of a massive body like the Earth, the Earth's stress energy gives rise to spacetime curvature such that the geodesics are all trajectories accelerating towards the Earth's center, at an acceleration $g$ on the Earth's surface.
However, non gravitational physics "messes this up" and "gets in the way". A body falling towards the center of the Earth can't do so for reasons of solid state physics: solid things like Earth surfaces and feet can't pass through one another. So an equilibrium is found where the Earth pushes back on the soles of our feet (or our bottoms and legs if we're sitting) so that we accelerate constantly upwards away from the geodesic motion to the tune of $g$ meters per square second acceleration.
But if we take these solid state physical processes away, by dropping a body of the edge of a table, say, then it will briefly undergo geodetic motion such that we, in our non inertial reference frame (stationary with respect to the Earth's surface), see the body undergoing an acceleration independent of its inertial mass.
A: Well Inertia and gravity are same at fundamental level.
Inertia - A body due to its mass (energy), creates a dip of space around it. That dip makes a force be required to make a change in state of the body. Hence causes inertia.
Gravity - Same dip (curve) due to mass (energy) of the body manifests as gravity for other bodies.
So, their origin is same and that is curving of space.
Inertia is nothing but gravity of the body acting on itself against any change of state. Therefore gravitational and inertial mass are same.
My view is that gravity and inertia are same phenomena. They are two sides of same coin.
Curving of space by mass/energy of a body manifests as gravity for other bodies.
Same curving of space manifests as inertia of the body, when we try to change its state of rest, or uniform motion.
I would be happy if someone busts this view conceptually, or mathematically.
A: Gravity and inertia are not the same.  Inertia is the “change” of the center of gravity.   If gravity and inertia are the same, then there is no difference between a fast ball and a curve ball!
