When referring to a three-dimensional plane, a z-axis refers to the depth of a three-dimensional object. In the picture to the right you can see the z-axis plane going front to back and intersecting with the y-axis and z-axis.
A good example of where the z-axis is used on a computer is in computer games or other 3-D environment where you have a game character or other object moving into or out of a 3-D world.
When referring to a three-dimensional plane, a z-axis refers to the depth of a three-dimensional object. In the picture to the right you can see the z-axis plane going front to back and intersecting with the y-axis and z-axis.