The Born approximation is when you technically have an extended body but you ignore the scattering from the object itself. An example would be if you had a material that was almost transparent, like a stained glass that is only ever so super lightly stained. You can basically treat each part as if it saw the normal unchanged light. Technically the parts in the middle are seeing a little bit different light since some was scattered by the layers closer to the edge. However, it is almost 100% transparent, so the light they see is almost entirely the same as the original incident light. The scattered light from the outer layers didn't change anything much.
The dipole approximation is when you assume that the scattering body is very, very small, so that the exponential in the scattered wave can be approximated very well by the lowest order terms, which contribute in a dipole fashion. This is a good approximation if the wavelength is much larger than the object, so the exponential is very uniform in the region where the scatterer is concentrated.