1. **Problem statement:** Define a great circle on the sphere $S^2$ and determine the number of intersection points of two distinct great circles $C_1$ and $C_2$ on $S^2$. 2. **Definition:** A great circle on the sphere $S^2$ is the intersection of $S^2$ with a plane passing through the center of the sphere. It is the largest possible circle on the sphere and divides the sphere into two equal hemispheres. 3. **Intersection of two great circles:** Two distinct great circles $C_1$ and $C_2$ correspond to two distinct planes through the center of the sphere. The intersection of these two planes is a line through the center. 4. **Number of intersection points:** The line of intersection of the two planes intersects the sphere $S^2$ in exactly two antipodal points. Therefore, two distinct great circles intersect in exactly two points. 5. **Explanation:** Since the sphere is symmetric, the two intersection points are opposite each other on the sphere, called antipodal points. **Final answer:** Two distinct great circles on $S^2$ intersect in exactly two points.