From "God's Equation - Einstein, Relativity and the Expanding Universe" by Amir D. Aczel

...the task remained to solve the equation. This, in a sense, was Einstein's challenge to the world of science - solve my equation and learn something about the laws of nature. Solving Einstein's field equation means finding the metric that satisfies the equation. It means determining the quantity ds² of the Riemann metric that applies to any situation as dictated by the equation.

Once one obtains the "line element" ds², one knows the shape, the curvature of spacetime in the particular situation to which Einstein's equation is applied. One knows how the "lines" in this curved space look and what constitutes the shortest path between two points. For example, if one applies Einstein's equation to a sphere, one finds that a shortest-distance "line" in this space is a great circle, as every airplane pilot or ship's navigator knows. The equation, of course, is applied to more complicated spaces where gravity plays a role. It accounts for how gravity itself imposes curvature on the space.

The process Einstein set in motion by proposing his equation continues to our own day - and intensifies all the time. Solutions of Einstein's equation have led us to discover fantastic phenomena that the equation predicts. These include gravitational waves, the warping of space, the phenomenon of "frame dragging," where space-time is whirled around a spinning massive body, not to mention the perihelion problem, the gravitational red shift, and the bending of light. But the first solution to Einstein's equation (except for his own solutions leading to the redshift, bending of light, and other effects Einstein determined on his own) was provided even before the theory was complete by a soldier in World War One.

On January 16, 1916, and again on February 24, 1916, Einstein read before the Prussian Academy two papers written by Karl Schwarzschild (1873-1916), a brilliant German astrophysicist and Director of the Potsdam Observatory, who was the first to solve Einstein's field equation of gravitation. Schwarzschild's solution later led to the understanding of black holes, and ultimately dictated the strong implications of Einstein's field equations on cosmology. Schwarzschild could not read his own papers in front of the academy, because he was at that time languishing in the trenches of the Eastern Front of World War I. There in the battlefields, facing the Russians, Schwarzschild had read Einstein's paper with the equations and solved them. He mailed the solution to Einstein in Berlin. Schwarzschild died on May 11, 1916, from a disease he contracted on the front. On June 29, Einstein read an obituary of Schwarzschild in front of the Prussian Academy.