the geodesic equation
What is a straight line when the space itself is curved? On a globe you cannot draw a truly straight line, but you can draw the straightest possible one: a great circle, the path an aircraft or a taut string naturally follows. Such a curve is a geodesic. In general relativity a body in free fall, feeling no force but gravity, coasts along a geodesic of curved spacetime, so a planet orbiting the Sun is doing nothing more exotic than following the straightest available path.
The geodesic equation is d^2 x^a / dtau^2 + Gamma^a_bc (dx^b/dtau)(dx^c/dtau) = 0, where tau is the proper time along the worldline and Gamma^a_bc are the Christoffel symbols built from the metric. The first term is ordinary acceleration; the second is the correction that accounts for the coordinates and the curvature. Equivalently, a timelike geodesic extremizes the proper time between two events, and for particles it is in fact the path of longest proper time, the deep reason a freely-falling clock reads more elapsed time than an accelerated one.
This equation replaces Newton's 'F = m a with a gravitational force'. There is no force term: gravity has been absorbed entirely into the Gamma symbols, that is, into the geometry. It is the mathematical statement of the equivalence principle, that free fall is unforced motion. One honest caveat: light and other massless particles also travel on geodesics, but proper time vanishes along their paths (dtau = 0), so you must parametrize them by an affine parameter rather than by tau.
The Moon does not orbit the Earth because a force reaches out and pulls it. In Einstein's picture the Earth's mass curves the surrounding spacetime, and the Moon simply coasts along the straightest path through that curved geometry, which happens to close into a near-circle.
Orbit as geodesic: unforced motion through curved spacetime.
The straightest line is not always the shortest. For timelike paths in spacetime the freely-falling worldline is the one of maximal proper time, a consequence of the metric's mixed signature. This is the geometric heart of the twin 'paradox'.