What exactly is the relationship between gravity and acceleration? I know that one of Einstein's key steps towards General Relativity was the realization that gravity and acceleration were related. I understand the free-falling elevator analogy, but the relationship between inertial and gravitational mass is still a bit ambiguous to me.
 A: Gravity causes acceleration, but acceleration can happen from a lot of other things as well, for example, on electromagnetic effects.
In most cases, the acceleration depends on some charge-like quantity. For example, a body with a mass of 1kg and with a charge of 1C will accelerate faster in the same electric field, as a body with 2kg of mass and with the same charge.
The only exception is the gravity. A body with 2kg of mass experience twice bigger force on gravity, as a body with 1kg. The result is that their acceleration is exactly the same. It was the result of Galilei's well known experiment (as he threw out bodies with different mass from the Pisa Tower).
No other interaction behaves similarly. To explain this, there are two possible ways:
1) We can say, that in the case of the gravitation, the mass behaves like some charge. It was Newton's concept.
2) We can say, that the gravitational acceleration is independent from the mass. Actually, it is independent from everything, except from the geometric configuration of the examined system.
Here comes Occam's razor in the picture. Occam's razor means, that explaining an unknown thing, the simplest explanation is the most probable.
This leaded to the key concept, that the gravity is not a charge-proportional, but only a purely geometric effect.
Now consider: there is an experimental condiguration, with many parameters, between them geometric parameters, mass distribution and charge-like parameters. Newton said, gravitation depends on all of them. Einstein's key concept was to minimize it, so he geometrized the laws.
Before that, everything is worked in the exactly opposite direction: there were the geometrical, experimental results, and essentially the science described these by algebraic formulas. Now Einstein stepped in the other direction, he geometrized the science again.
Finally, the GR is much more complex (in algebraic sense, too) as Newton's single-formula solution, but the theory is also much better.
Consider a sheet a paper, where the horizontal direction symbolizes the movement in space, and the vertical the movement in time. We are continually moving in the direction of the time, so we are always moving upwardly.
If you curve this paper, for example, if you slouch one of its corners, then the time-directional movement on this paper will also result a horizontal movement, a movement in the space. The masses give curveture to this paper, and thus they cause acceleration.
A: They are, as Einstein pointed out, equivalent. So why distinguish between the two? Well the only real difference is that they are measured differently. To measure inertial mass, we exert a given force to something with an unknown mass. To measure gravitational mass we compare the force of gravity from an object with an unknown mass to the gravitational force of a mass that we know. Whenever we compare our results using either method, however, we always arrive at the same result. This my friend is the equivalence principle!
Here's a link that explains nicely (also a major source of my info): http://www.physlink.com/Education/AskExperts/ae305.cfm
P.S. This is also the principle that fuels flat-earth conspiracies so don't let those darn flat-earthers fool you.
