What is the difference between intrinsic and extrinsic curvature?
Solution 1:
Intrinsic curvature comes from the parallel translation of a vector tangent to the path of translation. If a vector is translated around a loop and it fails to come back onto itself that is intrinsic curvature. Extrinsic curvature is computed by the parallel translation of a vector normal to a surface or space. If the translated normal deviates from the normal vector at a point the difference in the two normal vectors $\delta\bf n$ defines the extrinsic curvature $\delta\bf n~=~\bf K\delta e$ for $\delta e$ a unit of translation along the space.
An example of this is the torus. This is easily visualized with the 2-torus. The inner region appears curved in a saddle shape, while the outside is more spherically curved. The sum of the two is zero however. If I cut the torus into a square the geometry is similar to the old video games where Pacman leaves the top and emerges at the bottom, or exists the right and enters the left and visa versa. This square sheet is flat; it has zero intrinsic curvature. The curvature we see in a 2-torus as an inner tube is extrinsic due to its embedding in three dimensions.
Solution 2:
A Riemannian manifold is intrinsically curved if there exists a geodesic triangle bounding a topological disk whose interior angles do not add to $\pi$.
An embedded Riemannian submanifold $M \subset N$ is extrinsically curved if no smooth orthonormal frame of normal vectors of $M$ is parallel (convariantly constant) in $N$.
A flat square torus embeds isometrically as a product of circles in four-dimensional space; the image is intrinsically flat (every triangle bounding a disk has total interior angle $\pi$) but extrinsically curved (there is no parallel field of unit normal vectors, much less a parallel frame).
A great sphere in a round $3$-sphere is intrinsically curved (isometric to the unit sphere in Euclidean $3$-space, so a triangle bounding a disk has total interior angle $> \pi$) but extrinsically flat (each of the two smooth unit normal fields is parallel). If you don't mind a non-compact example, a circular cylinder in Euclidean $3$-space is intrinsically flat but extrinsically curved for similar reasons.
Gravitational lensing may be viewed as a physical manifestation of intrinsic curvature: In an approximation where the lensing system is static, the path of a light ray is a spatial geodesic. Two light rays seen at distinct points of the sky form a "digon", a geodesic polygon with two sides. Because the sum of the interior angles is not zero, space is intrinsically curved.