For example, in Newton's conception of gravity, the Earth orbits the Sun because the Sun exerts a gravitational force upon the Earth which pulls it inward and causes its motion to depart from a straight line. (The Earth also exerts a gravitational force upon the Sun, but because the Sun is so much more massive, this can be neglected to a first approximation.) In general relativity there is no gravitational force. The Earth is moving in a straight line in spacetime, but because the Sun curves spacetime in its vicinity this geodesic traces out a helix in spacetime which we perceive as the Earth's orbit.
Now, if this were a purely qualitative description, one could dismiss it as philosophical babble, but Einstein's theory provided a precise description of the gravitational field and the motion of objects within it and, when the field strength is strong or objects are moving very rapidly, makes different predictions than Newton's theory. In particular, Einstein's theory predicted that the perihelion of the orbit of Mercury would rotate around the Sun more rapidly than Newton's theory could account for, that light propagating near the limb of the Sun or other massive bodies would be bent through twice the angle Newton's theory predicted, and that light from the Sun or other massive stars would be red-shifted when observed from a distance. In due course all of these tests have been found to agree with the predictions of general relativity. The theory has since been put to many more precise tests and no discrepancy with experiment has been found. For a theory which is, once you get past the cumbersome mathematical notation in which it is expressed, simple and elegant, its implications are profound and still being explored a century later. Black holes, gravitational lensing, cosmology and the large-scale structure of the universe, gravitomagnetism, and gravitational radiation are all implicit in Einstein's equations, and exploring them are among the frontiers of science a century hence.
Unlike Einstein's original 1905 paper on special relativity, the 1915 paper, titled “Die Grundlage der allgemeinen Relativitätstheorie” (“The Foundation of General Relativity”) is famously difficult to comprehend and baffled many contemporary physicists when it was published. Almost half is a tutorial for physicists in Riemann's generalised multidimensional geometry and the tensor language in which it is expressed. The balance of the paper is written in this notation, which can be forbidding until one becomes comfortable with it.
That said, general relativity can be understood intuitively the same way Einstein began to think about it: through thought experiments. First, imagine a person in a stationary elevator in the Earth's gravitational field. If the elevator cable were cut, while the elevator was in free fall (and before the sudden stop), no experiment done within the elevator could distinguish between the state of free fall within Earth's gravity and being in deep space free of gravitational fields. (Conversely, no experiment done in a sufficiently small closed laboratory can distinguish it being in Earth's gravitational field from being in deep space accelerating under the influence of a rocket with the same acceleration as Earth's gravity.) (The “sufficiently small” qualifier is to eliminate the effects of tides, which we can neglect at this level.)
The second thought experiment is a bit more subtle. Imagine an observer at the centre of a stationary circular disc. If the observer uses rigid rods to measure the radius and circumference of the disc, he will find the circumference divided by the radius to be 2π, as expected from the Euclidean geometry of a plane. Now set the disc rotating and repeat the experiment. When the observer measures the radius, it will be as before, but at the circumference the measuring rod will be contracted due to its motion according to special relativity, and the circumference, measured by the rigid rod, will be seen to be larger. Now, when the circumference is divided by the radius, a ratio greater than 2π will be found, indicating that the space being measured is no longer Euclidean: it is curved. But the only difference between a stationary disc and one which is rotating is that the latter is in acceleration, and from the reasoning of the first thought experiment there is no difference between acceleration and gravity. Hence, gravity must bend spacetime and affect the paths of objects (geodesics) within it.
Now, it's one thing to have these kinds of insights, and quite another to puzzle out the details and make all of the mathematics work, and this process occupied Einstein for the decade between 1905 and 1915, with many blind alleys. He eventually came to understand that it was necessary to entirely discard the notion of any fixed space and time, and express the equations of physics in a way which was completely independent of any co-ordinate system. Only this permitted the metric structure of spacetime to be completely determined by the mass and energy within it.
This book contains a facsimile reproduction of Einstein's original manuscript, now in the collection of the Hebrew University of Jerusalem. The manuscript is in Einstein's handwriting which, if you read German, you'll have no difficulty reading. Einstein made many edits to the manuscript before submitting it for publication, and you can see them all here. Some of the hand-drawn figures in the manuscript have been cut out by the publisher to be sent to an illustrator for preparation of figures for the journal publication. Parallel to the manuscript, the editors describe the content and the historical evolution of the concepts discussed therein. There is a 36 page introduction which describes the background of the theory and Einstein's quest to discover it and the history of the manuscript. An afterword provides an overview of general relativity after Einstein and brief biographies of principal figures involved in the development and elaboration of the theory. The book concludes with a complete English translation of Einstein's two papers given in the manuscript.
This is not the book to read if you're interested in learning general relativity; over the last century there have been great advances in mathematical notation and pedagogy, and a modern text is the best resource. But, in this centennial year, this book allows you to go back to the source and understand the theory as Einstein presented it, after struggling for so many years to comprehend it. The supplemental material explains the structure of the paper, the essentials of the theory, and how Einstein came to develop it.