In both hex and square grid systems, we have a mapping of a two-dimensional plane. Because of this, any space in either system can be represented by coordinates that denote that space’s location relative to two non-parallel axes on that plane. The difference is, in a square grid those axes are typically perpendicular, while in a hex grid the axes are typically at a 60° angle.
We’ll call the space where the axes cross one another the Origin, and designate it (0,0). From here, it’s easy to see in a square grid how the coordinates are assigned. The number of spaces to the left or right of the origin is the x-coordinate, while the number of spaces above or below is the Y-coordinate.
Let’s focus on the x-coordinate for a moment. It’s easy to think of the x-coordinate as describing the current space’s distance from the y-axis, but that’s really only true in a square coordinate system, or any system with the two axes being perpendicular. Another way to look at it is that the x-coordinate identifies columns of spaces which run perpendicular to the x-axis.
Similarly, the y-coordinate denotes rows of spaces perpendicular to y-axis.
The end result is that we have coordinates that identify the row (y-coordinate) and column (x-coordinate) relative to the two axes individually. If we apply this to the hex system, we end up with a similar coordinate system, except the “rows” and “columns” are hexes rather than squares, and the axes are at a non-right angle.
You may notice that the “columns” and “rows” are adjacent hexes that are opposite one another. In this case, the x rows run upper-left to lower-right, while the y rows run upper-right to lower-left. There’s a third row that runs horizontally, which would denote a hex’s position relative to a vertical (z) axis.
This axis is totally unnecessary to uniquely identify spaces, but it can be convenient for computation and describing distances, relative positioning, etc. I will likely include this axis in most of my illustrations for completeness and for reference, but remember that only two axes are necessary to describe the location of a hex in a grid on a two-dimensional plane.