In geometry, a “great circle” is a flat plane that cuts through the middle of a sphere. Essentially, it becomes the longer line that was drawn around that sphere.
In aviation navigation, a “great circle route” is the shortest path between two points, and is achieved by finding the great circle that both points are on. This is invariably an arc, not a straight line.
Indeed, “straight” doesn’t mean much when talking about a sphere, because the entire surface is curved – nothing on a sphere is “straight.” When looking on a sphere, the great circle route actually looks fairly straight. It only appears to be a curved line or arc when flatten it to a Mercator projection map. (See image on this page.)
The great circle routes are why flights between the U.S. and Europe arc up and over the arctic circle, rather that going directly east or west.