A geodesic is the path of shortest distance between two points in a metric curved geometry. Because of the curvature, these distances are usually greater than the distance of a straight line between the two points. These geometries defy the common saying that "The shortest distance between two points is a line"; in fact, it's a geodesic.
Cartographers must account for geodesics when taking their measurements and surveying the land. Cartography is the modern form of geodesy, the study of measurements along the surface of the Earth. The Earth is a spherical geometry, the most common curved geometry.
Real World Examples
Epcot at Disney World is perhaps the most recognizable example of a geodesic. It is also spherical. See also geodesic domes.