A great circle, also known as a Riemannian circle, of a sphere is the intersection of the sphere and a plane which passes through the center point of the sphere, as distinct from a small circle. Any diameter of any great circle coincides with a diameter of the sphere, and therefore all great circles have the same circumference as each other, and have the same center as the sphere. A great circle is the largest circle that can be drawn on any given sphere. Every circle in Euclidean space is a great circle of exactly one sphere.
For any two points on the surface of a sphere there is a great circle through the two points. The minor arc of a great circle between two points is the shortest surface-path between them. In this sense the minor arc is analogous to “straight lines” in spherical geometry. The length of the minor arc of great circle is taken as the distance of two points on a surface of a sphere, namely great-circle distance. The great circles are the geodesics of the sphere.
When long distance aviation or nautical routes are drawn on a flat map (for instance, the Mercator projection), they often look curved. This is because they lie on great circles. A route that would appear as a straight line on the map would actually be longer.
No credit to me - thank Wikipedia - but I think the point is the bit in bold. This is the confusion to most folk.
I think the point to understand is that it is the "minor arc" of the great circle that presents the shortest route between two points. Other circles can be drawn, but will result in a longer minor arc (least that's what I understand).