In a cartesian coordinate system, the z-axis is the axis passing through $x = 0, y = 0$. which is perpendicular to both the x-axis and the y-axis.
It consists of all the points in the real vector space in question (usually $\R^3$) at which all the elements of its coordinates but $z$ are zero.
As the visual field is effectively two-dimensional, it is not possible to depict a three-dimensional space on a visual presentation (paper, screen etc.) directly.
Therefore the representation of the third axis of such a cartesian coordinate system is necessarily a compromise.
However, if we consider the plane of the visual field as being a representation of the x-y plane the z-axis can be imagined as coming "out of the page".
The usual convention for the orientation of the $z$-axis is that of the right-hand rule:
Let the coordinate axes be oriented as follows:
- Let the $x$-axis increase from West to East.
- Let the $y$-axis increase from South to North.
Then the $z$-axis increases from below to above.
If the $x$-axis and $y$-axis are aligned with a piece of paper or a screen aligned perpendicular to the line of sight, this translates into the following orientation:
- Let the $x$-axis increase from left to right.
- Let the $y$-axis increase from bottom to top.
Then the $z$-axis increases from behind to in front (that is, from further away to closer in).
The plural of axis is axes, which is pronounced ax-eez not ax-iz.