This distinction arises from the fact that once two perpendicular directions have been chosen for the x and y axes, there are two possible choices for the positive side of z.

The coordinate system x, y, z is called right-handed, if the three vectors are situated like the thumb, index finger and middle finger (pointing straight up from your palm) of your right hand. Alternatively, imagine you are gripping the z axis in a fist in such a way that your fingers curl round from x to y; the direction your thumb points in gives the positive z-axis of the right-handed system for the right hand; and likewise for left.

Another way of determining handedness of axes is as follows: imagine an ordinary screw lying along the z axis. Now turn it as if by a screwdriver, in the direction from x to y. The direction in which it moves, up or down, is the direction of the positive z axis in a right handed system. This applies for a screw that is tightened by turning it clockwise, with a so-called right-hand thread. This is the most common kind.

Left-handed on the left, right-handed on the right.