A geodesic is a generalisation of the notion of a straight line to curved spaces, and can be sometimes be thought of as the locally shortest or extremal path between points.
A geodesic is a generalization of a straight line to curved space. It is a length-minimizing curve, which is equivalent to a path that a particle which is not accelerating would follow.
On a Riemannian manifold, a geodesic coordinatized by coordinates $x^k$ satisfies the ordinary differential equation known as the geodesic equation: $$\frac{d^2x ^k}{dt^2}+\sum_{ij}\Gamma^{ij}_k\frac{d x^i}{dt}\frac{dx^j}{dt}=0.$$
In general relativity, a geodesic will describe the motion of point particles under the influence of gravity.
