a great circle
Slice a perfectly round orange straight through its centre, and the circular rim of the cut is a great circle. On a globe, the equator and every line of longitude are great circles; lines of latitude (except the equator) are not, because they do not pass through the Earth's centre. A great circle is the biggest circle you can possibly draw on a sphere.
Precisely: a great circle is the intersection of a sphere with a plane that passes through the sphere's centre. Such a circle has the same radius r and the same centre as the sphere itself, so it has the largest possible circumference (2·pi·r) and divides the sphere into two equal hemispheres. Any other plane cutting the sphere off-centre gives a smaller circle, called a small circle. A plane parallel to a great circle cuts the sphere into curved slices called zones.
Great circles are the sphere's 'straight lines': the shortest path between two points on a sphere always lies along the great circle through them — which is why airlines fly curved-looking great-circle routes, the genuinely shortest path over the globe. This makes a sphere a model of a non-Euclidean (spherical) geometry, where 'lines' are great circles and, strikingly, every pair of them meets — there are no parallels. The surface area of a sphere, 4·pi·r^2, equals exactly four great-circle areas (4 · pi · r^2).
On Earth (radius about 6371 km), the equator is a great circle of circumference 2·pi·6371 ≈ 40,030 km; a flight from one point to another follows the great circle through them, the shortest route over the surface.
A great circle passes through the sphere's centre and is its 'straightest' path.
Only circles through the sphere's centre are great circles; latitude lines other than the equator are small circles, and on a sphere any two great circles meet, so there are no parallel 'lines'.