what are the differences between orthogonal and non orthogonal coordinate systems ?

Answer 1

A coordinate system is orthogonal if all of the vectors that you use to define the coordinate system are orthogonal. Orthogonal means they meet at a right angle.

For instance the standard coordinate system for R^3 is i hat, j hat and k hat. These are all at right angles so this is an orthogonal coordinate system

A nonorthogonal coordinate system will be one in which not all the vectors are at right angles.

Answer 2

This example will make it clear for you.
Think of the usual two dimensional system we use for graphing: cartesian coordinate system which has a horizontal and a vertical axes. These are orthogonal to each other meaning that they are perpendicular or meet at 90 degrees angle.
The two dimensional system is non-orthogonal if these axes meet at a different angle.

Now, if we have more than two dimensions, we say that the coordinate system is orthogonal if its axes are prependicular. Otherwise, it is called non-orthogonal.