It is normally perpendicular to both of them - passing through the intersection of those axes - i.e. through the point (0,0,0)
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For an example, if negative and positive directions on the X and Y axes represent left/right and forward/backward, Z axis represents up/down.
The x axis is horizontal and the y axis is vertical.
A point on a 3-d coordinate system would take the form of (x,y,z). You go x units on the x-axis (left or right), y units on the y-axis (up or down), and z units on the z-axis, (front or back).
The side is the WRONG answer. A Y axis is up and down and a X axis is left and right. Easiest way to remember is Y to the sky, X to the side. Also, the Z axis is front to back.
In 2-dimensions, it is the y-axis. In 3-dimensions, though, the x and y axes form the horizontal plane and the z-axis is the vertical axis.