Newtonian gravity predicts closed circular and elliptical orbits. This prediction depends on the fact that Newtonian gravitational force is directed along the line joining the instantaneous positions of objects, like the Earth and the Sun. If Newtonian gravitational force weren't directed along this line, orbits wouldn't be closed.
As the Earth orbits the Sun, the position of the Sun, relative to Earth, changes. If gravity propagates at the speed of light, shouldn't the Earth feel (gravitationally) where the Sun was (according to the Earth) eight minutes ago, that is, shouldn't gravitational force be directed along the line that joins where the Earth is now to where the Sun was eight minutes ago? And if this is true, then, according to the previous paragraph, how can the Earth's orbit be a closed ellipse?
To answer these questions, I am going to talk briefly about the main equation of Einstein's theory of gravity, general relativity, G = T. Here, G is a geometrical quantity that depends on the curvature of spacetime, and T is a physical quantity that depends on the distribution and flows of mass and energy in the universve.
In Einstein's theory, gravity is a manifestation of spacetime curvature. If T depends not only on position, but also on flow of matter, then (by the equals sign) G, spacetime curvature, and (thus) gravity are affected by the velocities of objects. This feature is not present in Newtonian gravity.
As an example, consider a uniformly dense planet. According to Newton, the gravitational field of the planet is independent of the spin of the planet. According to Einstein, however, a planet's gravitational field is not independent of its spin. Spin puts the matter of the planet in motion, so different spins give different gravitational fields. To test this for the Earth, a satellite carrying gyroscopes has been put into orbit.
Back to the Earth and Sun. Form the point of view of the Earth, the mass of the Sun moves, and so, according to Einstein, this motion contributes to the gravitational field of the Sun. The field of the Sun depends on where the Sun is, and on how the Sun moves.
These two contribution's to the Sun's gravitational field, position and velocity, add to produce an "effective force" that *appears* to point towards where the Sun is now, not where it was eight minutes ago.
What happens if the Sun magically disappears? The Earth will continue on in its orbit for another eight minute under the influence of an "apparent force" directed towards where the Sun would have been. After eight minutes, the Earth realizes that the Sun isn't there, and stops orbiting the missing Sun.
