A geodesic is the natural, unforced path an object follows through spacetime when nothing but gravity is acting on it. On a flat surface, a geodesic is just a straight line, but on a curved surface, like the surface of the Earth or the fabric of spacetime, a geodesic can curve while still being the most direct route available.
This is the core idea that replaces the old picture of gravity as a force pulling objects around. A planet orbiting the sun is not being yanked into a circle, it is simply following a geodesic through spacetime that has been curved by the sun’s mass, and that curved geodesic happens to loop back on itself.
Even the straight fall of a dropped object is a geodesic, the object is following the straightest path available through spacetime near the Earth, it just happens that this path curves toward the ground because of how the Earth’s mass bends spacetime nearby.
Mathematically, a geodesic is defined by a specific equation, the geodesic equation, which describes a path along which a moving object’s velocity is parallel transported, meaning it changes only in the way the curved space itself demands and not through any external push. On a flat plane, parallel transport keeps a vector pointing in exactly the same direction the whole way, which is why geodesics there are ordinary straight lines. On a curved surface, like the surface of a sphere, parallel transporting a vector around a loop can leave it pointing in a different direction than where it started, a visible sign of curvature. That same mathematical machinery, generalized to four dimensional spacetime, is what lets physicists calculate the exact orbit a planet or a light ray will trace around a massive object.