Short answers, pulled from the story.
A geodesic in general relativity is the generalization of a straight line to curved spacetime. The world line of a particle free from all external, non-gravitational forces follows a geodesic, meaning freely moving and freely falling particles always travel along geodesics.
In general relativity, gravity is not a force but a consequence of curved spacetime, where the curvature is sourced by the stress-energy tensor. A planet's orbit is the projection of a four-dimensional geodesic in the curved spacetime around a star onto three-dimensional space.
Christoffel symbols, also called affine connection coefficients or Levi-Civita connection coefficients, appear in the geodesic equation and are symmetric in their two lower indices. They are functions of the four spacetime coordinates only and do not depend on the velocity or acceleration of the particle.
Physicist Steven Weinberg derived the geodesic equation directly from the equivalence principle. He began by assuming that a freely falling particle does not accelerate relative to a freely falling coordinate system, then applied the multi-dimensional chain rule and defined the affine connection to arrive at the geodesic equation.
Einstein believed the geodesic equation should follow from the field equations for empty space, specifically from the vanishing of the Ricci curvature, rather than being an independent postulate. However, David Malament later argued that the geodesic principle is not a consequence of Einstein's equation alone and that additional assumptions are required.
Massive particles follow timelike geodesics, while massless particles such as photons follow null geodesics. The distinction is encoded in the geodesic equation by replacing the negative-one value used for timelike geodesics with zero for null geodesics.