Essentially the "straightest path" in a curved space or curved spacetime. This is the path followed by an object with no forces acting on it. In the curved spacetime of General Relativity, these paths may seem to be very curved — even appearing as circles or ellipses, for example. A geodesic is easily understood by looking at a very small region around the object. Even in highly curved spacetime, a small enough region will seem flat, so there is a natural idea of a "straight path". By following short segments, the whole geodesic is built up into one long path.
Now, here, you see, it takes all the running you can do, to keep in the same place. If you want to get somewhere else, you must run twice as fast as that.
The Red Queen in Lewis Carroll's Through the Looking Glass