Occurring to Albert Einstein in 1907, the Equivalence Principle can most easily be stated that if one's in free fall, one won't be able to feel one's own weight. Restated, this means that in a freely falling reference frame, locally (for sufficiently small changes in time and space) it appears like a gravity and acceleration free reference frame.
To quote Einstein:
I was sitting in a chair in the patent office in Berne when all of the sudden a thought occurred to me: If a person falls freely he will not feel his own weight. I was startled. This simple thought made a deep impression on me. It impelled me toward a theory of gravitation.
This has many and various implications, both for Newtonian Mechanics and for Einsteinian Physics. Newtonian implications include that the Gravitational Mass and the Inertial Mass (the value of mass that gravity acts on and the M in the venerable F=MA) are indeed one and the same. One could argue that "the mass is the mass! Why does this need to be shown?" Well, gravity is still one of those spooky forces that we don't have a good understanding of, and it's only really out of experience and experiment that we know that inertial mass and gravitational mass are the same; there's nothing forcing those masses to be the same (there could be alternate universes where they are not).
This implies the Universality of Free Fall, the concept that all objects will accelerate at the same speed in free fall (on Earth, around sea level, the value is about 9.8 meters/second2 (No, I will not give this in the imperial system, real people don't use that anymore)).
As for Einsteinian implications, it gets a little more complicated. Take a freely falling elevator falling towards the Earth, with an observer inside, and two apples on either side of the observer. By the Newtonian implications, there's no local difference between this frame and one that's not accelerating or affected by gravity. However, if you look at this frame for long enough, the distance between the apples will get smaller, since they're both on trajectories straight towards the center of the earth. Since in the reference frame of the elevator, there's been no force on the apples, this implies that spacetime itself must be curved by masses (in our case, the Earth).
The paths of objects that aren't acted on by any net force define our notion of a straight path, and a straight path can be curved by curved space-time (this segues nicely into Einstein's General Theory of Relativity. This also implies that there can be no measurement apparatus entirely contained within the closed reference frame (elevator) that can tell the difference between a gravitational acceleration and an inertial acceleration (or lack of either) locally.
I should take a minute to explain this "locally" business. It is kind of a cop-out; the simple apparatus of the two apples can detect the difference between an elevator falling in gravity and one out in deep space with no gravity! The point is this: for very small distances and changes in time, there is no difference between the frames, and since there's no difference between them there, logically they must be the same thing, but with something else interfering in the one case, and that's the curving of spacetime.
Sources: Resnick and Halliday's Physics Volume 1 5th edition, http://www.npl.washington.edu/eotwash/equiv.html, and http://csep10.phys.utk.edu/astr162/lect/cosmology/equivalence.html