A ZigZag structure is an undirected graph. Nodes are arranged in an n-dimensional space. For each coordinate axis, every node can only have one edge in the plus-1-along-coordinate direction, and one edge in the minus-1-along-the-coordinate. I.e. no diagonal connections, and max one connection in each "cardinal direction" (+ and - in n dimensions = 2n directions).
In its simplest form, a ZigZag structure can be an n-dimensional grid, where each node is connected to its direct neighbors in the grid. In 2D, that would give you a spreadsheet. However, ZigZag can do more: you can have loops along any axis, e.g. a spreadsheet where the columns "wrap around". Furthermore, ZigZag supports looping the columns of only one particular row: a normal 2D spreadsheet, except that row 13 only has 4 columns, and those columns are wrapped up in a loop. You could also have sparse grids - where nodes are missing, and the edges skip over the missing node.
The idea is to have something that locally, if you look only at the surrounding nodes, always looks like a neat n-dimensional grid. But the global topology can be totally whacky.
I have no idea what you would use such a thing for.