Backing and veering are just a change in wind direction. These changes, as with most wind changes, are usually a result of pressure changes or fronts and air masses moving and pushing air around.
A veering wind is a wind that changes direction clockwise, thus veering to the right such as turning from west to north.
A backing wind is the opposite of a veering wind. This is when the winds change direction counter clockwise, backing to the left such turning south to east.