I have been told that the only straight lines on a sphere are great circles. Great circles are circles which pass through the "equator" of the circle. Why are these considered straight? They certainly look curved, because they are on a sphere.
Straight isn't a good description. However, the shortest path between any two points on a sphere lies on a great circle just like in flat space, the shortest path between any two points lies on a straight line.
They are as straight as they can be, given that they have to lie on a sphere. Lines in Euclidean space and great circles on spheres are examples of what's called geodesics in differential geometry.