The shortest path between two points on the surface of a sphere lies along a great circle. On a 2-dimensional map, this looks like a line, but when it's on a 3-dimensional sphere, it's an arc... part of a circle.

Full text in animation:

The shortest distance between two points on a sphere is a great circle, or a circle whose plane passes through the center of the sphere. Longitude lines are special great circles that pass through the north and south poles and whose planes pass through the center of the Earth. Latitude lines are small circles, that is, their planes do not pass through the center of the Earth.

The Equator is special, however, and does pass the through the center of the Earth, so it is a great circle too. The great circle path, however, is not always intuitive since we are used to looking at maps of the earth where sphere is represented on a flat surface.

For example, Moscow is located at about 55 N latitude. Imagine an earthquake near Japan at 34 N latitude, south of Moscow. It would appear that the shortest path from Moscow to the earthquake heads SOUTHeast, but for distances this large, we cannot trust a flat representation of the globe. The shortest distance from Moscow to the earthquake is on a Great Circle that actually heads NORTHeast out of Moscow.

If you're flying non-stop from New York to Tokyo, even though you start at about 40 N latitude and end up at 35 N latitude, SOUTH of your starting point… your flight heads NORTHwest out of New York, crosses Alaska en-route at about 70 N latitude, and ends going SOUTHwest into Tokyo.

The shortest distance between any two points on the earth is always along a great circle path.