The equality of inertial and gravitational mass has been recognized since at least the time of Galileo, and more recently has been empirically verified to 1 part in 1012. As a consequence of this equality, the free fall path of a small test particle in a gravitational field is independent of the particle's mass and composition. This fact is sometimes called the "weak equivalence principle". Einstein extrapolated this idea into what is called the "strong equivalence principle", which asserts that gravity is nothing other than the effects of the geometry of spacetime. In some modern popular accounts of general relativity the strong equivalence principle is expressed in terms of a supposed impossibility of distinguishing between inertia and gravity inside an elevator car (for example). Although these descriptions are evocative and somewhat useful for conveying the underlying idea of the strong equivalence principle - which is one of the most profound and significant in all of physics - they sometimes cause people to misconstrue the essential meaning of the principle, and to go off trying to think of ways to distinguish between various kinds of acceleration fields and gravitational fields. For example, they may cite the tidal effects of a non-uniform gravitational field as a means of distinguishing the effects of such a field from the effects of linear acceleration, thereby (they imagine) contradicting the popular descriptions of the strong equivalence principle.
Such objections fundamentally miss the point of the equivalence principle, which refers to the intrinsic rather than extrinsic qualities of gravity and acceleration. To make this clear, it's important to point out that the principle of "indistinguishability" is valid (up to some finite level of precision) only locally, which is to say, only over a sufficiently small interval of both space and time. Unfortunately, talking about "levels of precision" and "sufficiently small intervals" sometimes causes people to think that the equivalence principle is, in some sense, only approximately valid, and so they may be tempted to underestimate its significance. Of course, it's undeniably true that many of the popular expressions of the equivalence principle are only approximately true, but this shouldn't cause us to lose sight of the fact that the actual equivalence principle is precisely valid - at least as far as we can verify with current experimental techniques.
It's worth noting that, even the "weak equivalence principle", i.e., the equality of inertial and gravitational mass, is an intrinsic principle that applies only on a limiting sense. The accelerations used in establishing this correspondence can be defined with great precision only when the test objects are sufficiently small, so that they don't contribute significantly to the gravitational field in the region of interest, and therefore don't "act back" significantly on the gravitating body and upset our frame of reference. Thus, even neglecting air resistance and the rotation of the Earth, we cannot say that every object will take the same amount of time to "fall to Earth" from a given height, because that is only strictly true in the limit of infinitesimally small test particles. A real falling massive object contributes its own gravity, and actually pulls the Earth toward it, reducing the time it takes for the system to collapse.
Nevertheless, this doesn't change the fact that inertial and gravitational mass are (as far as we know) identical, because the equivalence is understood to be intrinsic. The "force" of gravity couples to all infinitesimal particles of mass in the same way, which is what makes it feasible to interpret gravity as the effect of spacetime geometry. The equivalence is exact (as far as we can tell), and this is a very profound and significant fact about the physical world, and it applies at each and every point in spacetime, but in the case of extended objects we must take account of the non-gravitational forces that are holding the particles of mass in fixed configurations, rather than allowing them to each follow geodesic paths. Thus, the equivalence principle relies on our ability to conceptually separate out gravity from the other forces of nature, which are never entirely absent, so this again involves an act of abstraction to isolate the equivalence principle in its pure and exact sense.
In developing the general theory of relativity, Einstein found that, beginning from a Minkowskian framework, the field equations of gravity follow almost uniquely from the combination of the equivalence principle and the requirement of optimal simplicity under general covariance. Thus the field equations of general relativity are virtually the embodiment of the equivalence principle. In this theory, the influence of both inertia and gravitation on a particle according to the general theory of relativity is expressed by the equations of motion
It's interesting that Einstein consistently emphasized that the Christoffel symbols G mab represent the gravitational field, in spite of the fact that this sort of comment would earn most students a failing grade in a university course on relativity. The "correct" view is that the Riemann curvature tensor Rmnab represents "true" gravity, whereas the Christoffel symbols can be non-zero in perfectly flat spacetime, simply by virtue of curvilinear coordinates. To illustrate, consider a flat plane with either Cartesian coordinates x,y or polar coordinates r,q as shown below
With respect to the Cartesian coordinates we have the familiar Pythagorean line element (ds)2 = (dx)2 + (dy)2. Also, we know the polar coordinates are related to the Cartesian coordinates by the equations x = r cos(q ) and y = r sin(q ), so we can evaluate the differentials
which of course are the transformation equations for the covariant metric tensor. Substituting these differentials into the Pythagorean metric equation, we have the metric for polar coordinates (ds)2 = (dr)2 + r2 (dq )2. Therefore, the covariant and contravariant metric tensors for these polar coordinates are
and we have the determinant g = r2. The only non-zero partial derivatives of the covariant metric components are 
and 
, so the only non-zero Christoffel symbols are G rqq = -r and G qqr = G qrq = 1/r. Inserting these values into (1) gives the geodesic equations for this surface
Since we know this surface is a flat plane, the geodesic curves must be simply straight lines, and indeed it's clear from these equations that any purely radial path (for which dq/ds = 0) is a geodesic. However, paths going "straight" in the q direction (at constant r) are not geodesics, and these equations describe how the coordinates must vary along any given trajectory in order to maintain a geodesic path on the plane. Of course, if we insert these polar metric components into Gauss's curvature formula we get K = 0, consistent with the fact that the surface is flat. The reason the geodesics on this surface are not simple linear functions of the coordinates is not because the geodesics are curved, but because the coordinates are curved.
However, if some "flatlanders" living on a small region of this plane were under the impression that the constant-r and constant-q loci were straight, then they might well imagine that the geodesic paths were curved, and that objects which naturally follow those paths are being influenced by some "force field". This may seem far fetched, but suppose we are standing in an upwardly accelerating elevator. We will tend to view things with respect to a coordinate system co-moving with the elevator, and as a result we notice that the natural paths of things are different than they would normally be, as if those objects were being influenced by an additional force field. This is exactly analogous to the flatlanders, except that it is their q axis which is non-linear, whereas our elevator's t axis is non-linear. Inside the accelerating elevator the additional tendency for geodesic paths to "veer off" is not really due to any extra non-linearity of the geodesics, it's due to the non-linearity of the elevator's coordinate system. Hence most people today would say that non-zero Christoffel symbols, by themselves, should not be regarded as indicative of the presence of "true" gravity. If the intrinsic curvature is zero, then non-vanishing Christoffel symbols simply represent the necessary compensation for non-linear coordinates, so, at most (the argument goes) they represent "pseudo-gravity" in such circumstances.
Despite all this, I'm inclined to think that Einstein had a valid point that hasn't been fully appreciated by all the subsequent writers of relativity text books. In a letter to his friend Max von Laue in 1950 he tried to explain:
...what characterizes the existence of a gravitational field from the empirical standpoint is the non-vanishing of theG lik, not the non-vanishing of the Riklm. If one does not think intuitively in such a way, one cannot grasp why something like a curvature should have anything at all to do with gravitation. In any case, no reasonable person would have hit upon such a thing. The key for the understanding of the equality of inertial and gravitational mass is missing.
The point of the equivalence principle is that, curving coordinates are gravitation, and there is no intrinsic ontological difference between gravity and pseudo-gravity. On a purely local (infinitesimal) basis, the phenomena of gravity and acceleration were, in Einstein's view, quite analogous to the electric and magnetic fields in the context of special relativity, i.e., they are two ways of looking at (or interpreting) the same thing, in terms of different coordinate systems. Now, it can be argued that there are clear physical differences between electricity and magnetism (e.g., no magnetic monopoles) and how they are "produced" by elementary particle "sources", but one of the keys to the success of special relativity was that it unified the electric and magnetic fields in free space without getting bogged down (as Lorentz did) in trying to fathom the ultimate constituency of elementary charged particles, etc.
Likewise, the most compelling aspect of general relativity is the "left-hand side of the equation" Rmn - (1/2)gmnR = Tmn (along with equation (1)), which unifies gravity and non-linear coordinates - including acceleration and polar coordinates - in free space, without getting bogged down in the "source" side of the equation, i.e., the fundamental nature of how gravity is ultimately "produced", why the elementary massive particles have the masses they have, and so on. What Einstein was describing to von Laue was the conceptual necessity of identifying the purely geometrical effects of non-inertial coordinates with the physical phenomenon of gravitation. In contrast, the importance and conceptual significance of the curvature (as opposed to the connection) is mainly due to the fact that it defines the mode of coupling of the coordinates with the "source" side of the equation.
Nevertheless, some authors make a point of saying that "gravitational effects are not equivalent to the effects arising from an observer's acceleration...", and they go on to describe how the effects of rotation (for example) can easily be distinguished from typical gravitation produced by massive bodies. In one sense this is obviously true, but on a deeper level it confuses intrinsic and extrinsic qualities by invoking two different limits when imposing the locality condition. Essentially these authors are correctly noting that what is a "sufficiently small" (i.e., local) region of spacetime for transforming away the translatory motion of an object to some degree of approximation may not be sufficiently small for transforming away the rotational motion to the same degree of accuracy, but this does not conflict with the equivalence principle; it just means that for an infinitesimal particle in a rotating body the "sufficiently small" region of spacetime is generally much smaller than for a particle in a non-rotating body, because it must be limited to a small arc of angular travel. Likewise some authors argue that the existence of tidal effects and geodesic deviation conflict with the equivalence principle, but that too is clearly not the case, provided we properly restrict our attention to a sufficiently small region of space and time.
Fundamentally, those who argue against the validity or meaningfulness of the equivalence principle are focusing on the wrong level of attributes when they point out that gravity and acceleration are generally distinguishable. In making this distinction they focus on one or more extrinsic attribute, whereas the equivalence principle is really an assertion of intrinsic equivalence. Indeed, it is the perfect intrinsic equivalence that makes a purely geometrical interpretation of gravity possible. The meaning of the equivalence principle is that the spacetime in a region with a gravitational field is intrinsically the same "stuff" at each point - up to a change of coordinates - as the spacetime in a region without a gravitational field. There is no additional coupling present to produce the effects of gravity on a test body. Gravity is geometry. This is expressed somewhat informally by saying that if we take sufficiently small pieces of curved and flat spacetime we can't tell one from the other, because they are the same stuff. What Einstein called "the happiest thought of my life" was his realization that the perfect equivalence between gravitational and inertial mass noted by Galileo implies that kinematic acceleration and the acceleration of gravity are intrinsically identical, and that this makes possible a purely geometrical interpretation of gravity.