A geodesic is a generalization of a straight line, a "shortest possible path", which in flat Euclidean space is indeed straight, but in curved spaces (or surfaces) that is less clear. The concept is used for the shortest paths on the surface of the Earth and for such paths within the curved space of general relativity. In the former case, the fact that the Earth is not a perfect sphere adds complication and using an oblate spheroid as the shape of the surface allows for more accuracy.
In GR, the path of a particle (or electromagnetic radiation) when no accelerating force other than gravity is present is a geodesic. GR treats gravity by conceiving space and time as a spacetime such that the effects of gravity are due to geometry, i.e., an object affected by gravity and otherwise uninfluenced is following a geodesic.
Two points in spacetime (i.e., each an instantaneous event, each at some point in space) can be separated in a time-like manner, i.e., a geodesic exists that could be traveled, or in a space-like manner, i.e., too far apart to be reached in the given time difference since it would require superluminal motion (faster-than-light travel).