What is difference between along the edge and across the edge
This is illustrated for a step edge. If you look at gray-scale values, one often considers that a 1D edge is (relatively) irregular across the edge (from gray to black), and regular along the edge (parallel to the border, the luminance value does not vary). 



You can perceive it better in the 3D view, where the amplitude along with the gray value is represented in 3D. 

In more generic cases, at the appropriate scales, the structure tensor is supposed to capture this behavior.