Why did we expect gravitational mass and inertial mass to be different? I've read many times that the fact that gravitational mass is equal to inertial mass (as far as we can tell) used to be a puzzle. I believe that Einstein explained this by showing that gravity is itself just an inertial force.
When I first encountered this concept, I thought "isn't there just one property called $m$ and it just appears in different equations (e.g. Newton's second law and the law of gravitation)? In a similar way that (say) frequency appears in many different equations."
Obviously I am thinking about this in the wrong way, but does anyone have a good way to explain why so that I can understand it?
 A: Perhaps one starting point for thinking about the equivalence principle between gravitational mass and inertial mass is the example of an object falling towards the Earth. Here, we know from Newtonian mechanics that $mg = ma$ implies that $a = g$, that is, the acceleration due to gravity is the same regardless of the mass of the object. Even though it may seem obvious today, it wasn't always clear whether the two masses in this equation were the same and whether two objects of different masses would have equal acceleration due to gravity. So, there have been increasingly precise experimental tests of the equivalence principle, even to this day.
Einstein developed a more complete framework for the equivalence principle, building on the ideas of Newton, Galileo and others. For example, consider the following elevator thought experiment. Suppose you are in an elevator with no windows, and you feel some force anchoring you against the floor of the elevator. You are unable to tell whether the elevator is on Earth where you are feeling the acceleration due to gravity, or whether the elevator is in a rocket in space accelerating at $1g$. This thought experiment represents an equivalence between a gravitational field and an accelerating reference frame.
A: 
"isn't there just one property called m and it just appears in
different equations (e.g. Newton's second law and the law of
gravitation)? In a similar way that (say) frequency appears in many
different equations."

There IS indeed just one property called m which appears in both the equations.
The point is that there is no intuitive reason why this should be the case.
Forget the term mass for a second and just think in terms of the properties of an object. One property of an object determines how strong is the gravity of the object.
The other property determines how much acceleration it experiences under a given force.
There is no obvious reason why these two properties should be the same.
But, we observe in daily life, that these two ARE the same.
That is what Einstein was able to explain i.e. why these two are the same.
EDIT: A good example to compare and contrast is to think about the forces between 2 electrically charged objects, as pointed out by Arthur's answer to this question. One property of the object (namely the charge) determines the amount of attractive/repulsive force. There is no reason why this property that determines the magnitude of a force would be the same as the property that determines how the object would move under a given force. And indeed these properties are not the same.
But in case of gravity, we observe, that these properties are the same.
A: Not sure if Einstein thought this way, but imagine you want to create a special relativity version of gravity, meaning you want to introduce "gravitational field" and construct evolution equations for the field. Newtonian gravity looks similar to electrostatics, forces are proportional to $1/r^2$, so you would think there should be similar stuff like "gravitational charge", "gravitational magnetic field" (at higher speeds) and etc. Given the analogy, it is indeed surprising that "gravitational charge"  is strictly proportional to the inertial mass, unlike in electricity, where you can have different masses for the same charge particles.
A: In addition to other good answers, the equivalence of inertial and gravitational masses is equivalent to the experimental fact that all masses fall at the same speed. Science knows that fact to be true since Galileo, and it seems obvious to us because we learned elementary physics long ago, but without doing the experiment it's actually far from obvious that a wood ball and an iron ball fall with the same acceleration. Therefore, until confirmed experimentally, it's far from obvious that inertial and gravitational masses are equal.
A: Objects have a property called "electric charge". This electric charge decides how strong a force they feel when close to other electrically charged objects. The electric charge of an object is more or less independent of inertial mass. So given a large, fixed, electrically charged object, you can make a small electrically charged test object feel different forces, and thus feel different accelerations by changing its electric charge without changing its inertial mass or the distance between the two.
Objects also have a property called "gravitational charge" (we call it gravitational mass). This gravitational charge decides how strong a force they feel when close to other gravitationally charged objects. The gravitational charge of an object could, in theory, have been independent of inertial mass. So given a large, fixed, gravitationally charged object, you could have a small gravitationally charged test object feel different forces, and thus feel different accelerations by changing its gravitational charge without changing its inertial mass or the distance between the two.
However, as far as we can tell, that's impossible. Can't be done. There is no inherent theoretical reason for why this can't be done. We can conceive of universes (or at least physical models) where it's entirely possible. Just model gravity after the electric force. But any experiment ever done points toward this being an impossibility.
Since it's impossible to separate the two properties, the physical thing to do is to go with the flow, listen to what our universe seems to tell us, and declare that they must actually be the exact same property.
A: Considering only the two equations: one is the Newton's second law $F = ma$, the other is the gravitational law $F = \frac{Gm_1 m_2}{r^2}$. The second become $F = mg$ close to the surface. These, in principle, are two different law and then, forgetting for a moment the names, we can use $d$ instead of $m$ in $F = dg$. We can say that $d$ is a property that quantify the attraction of the bodies, while $m$ in Newton's second law is a measure of the inertia, or the tendency of a body to stay in his state of motion. Then in principle, there is no reason for the two quantity to be equal. I leave it to other answers for a deeper insight, I just want to point out that, pragmatically, if we measure the ratio $\frac{m}{d} $ and this is equal to a constant compatible within experimental error, we can say they are the same thing. This ratio is equal to one under the right choice of units.
A: Gravitation is not a force. If you stand on the Earth then you are accelerated upward by the electromagnetic force. There is no force pulling you towards
the Earth. This upward acceleration is what makes you see your weight when standing on a scale.
On a heavier planet your weight increases. But by scaling the scale you always get the same value for your mass, which is nothing but a value assigned to inertia. This value is the measure of resistance to being accelerated.
Einstein realized that standing on a scale on Earth is equivalent to standing on a scale in a uniformly accelerated reference frame (the famous elevator thought experiment which Einstein used to demonstrate the equivalence principle) in outer space, the acceleration being equal to the acceleration on Earth (which, again, is directed upward).
This means that gravitational mass (the mass you can see standing on a scale on Earth) must be the same as inertial mass (the mass you can see while standing on a scale in a rocket that is accelerated in outer space with the value g).
In both cases it's the electromagnetic force (a true force, contrary to gravity) you experience. Also the other two true forces (the weak force and the color force) can be involved in acceleration, though they have a very short range.
A: Truly excellent question. All "why would we expect" questions are inherently subjective, and there is no single correct answer to this question. But I'm going to give a very heterodox (and I'm sure unpopular) answer: I would argue that you are correct, and in fact there never was any reason to expect that every object would have one inertial mass and a different gravitational mass. Nor was there any reason to expect that they'd be equal.
The currently accepted answer says "One property of an object determines how strong is the gravity of the object. The other property determines how much acceleration it experiences under a given force. There is no obvious reason why these two properties should be the same." If we're working entirely from empirics-free expectations, then I would disagree with this. There's no a priori reason to expect that any object has a single amount of "acceleration it experiences under a given force". Even if you assume that the acceleration must always be parallel to an applied force, you could certainly imagine a world where the proportionality constant varies depending on the type of applied force, e.g. its source. The fact that there is only a single "inertial mass" for all applied forces is already highly surprising, and it was not really fully appreciated until Newton.
Physics is all about unity, loosely defined as meaning "things that you wouldn't necessarily expect to be the same turn out to in fact be the same" (one clear example being symmetries of dynamical systems). You could certainly imagine a very messy world where every object has many different "inertial masses" for different types of applied forces. It's a remarkable empirical fact that all of these inertial masses turn out to be precisely equal. You could imagine that this remarkable unity might or might not extend even further to sourcing gravity; I personally don't find it particularly more likely that it stops before that than continuing even further.
I agree that after the formulation of electromagnetism in the 19th century, in retrospect it became more natural to consider by analogy the possibly of a "gravitational mass" that differed from the intertial one. But during the ~200 years between the two theories' formulation, I don't think there was really any particular reason to consider the possibility that the two quantities would be different.
A: Classical mechanics has a more or less axiomatic framework. Here, gravitational and inertial mass is represented by the same concept and the same symbol, $m$.
If we decide to disambiguate the concepts by representing them by different symbols, say $m$ for inertial mass and $M$ for gravitational mass and also by different concepts of mass, the resulting mechanics will still be consistent.
This is one way of characterising the difference between the two concepts of mass in mechanics.
A: The answer to your question is stress-energy. Both gravity and inertia are rooted in stress-energy.
In the case of gravity, it is more obvious, because we have general relativity, and one of its building blocks is that stress-energy is the cause of gravity, and everything and anything we know of that does possess stress-energy, does bend spacetime.

yes they do, and for the reasons you sketched out.

Do photons have inertia?
In the case of inertia the source being stress-energy is not so obvious. There is a nice example of this in the fact that even massless photons do have inertia. This is because they do have stress-energy, and so they do have inertia.

Symmetry of the Lagrangian with respect to translation in time and space (in classical mechanics), leading to conservation of energy and momentum.
The claim that the worldline of an object in free fall is a timelike geodesic of spacetime. (Such a worldline can also be described as a line of maximal proper time between any given pair of events on the line.)

Is there still no known origin of the law of inertia?
There are different ways to describe the origin of inertia, but most on this site agree that it can be deduced from conservation of energy and momentum, and geodesic motion.
The answer to your question is that since photons do have stress-energy, and this causes them to both bend spacetime and have inertia at the same time, proves (or gives us a hint) that both gravity and inertia are rooted (even if in the case of inertia the connection is non-trivial) in stress-energy. This could lead us to the expectation that gravitational mass and inertial mass should be equivalent. And then this is experimentally (based on the equivalence principle) proven.
