Question

A vector is directed to the right of the XZ-plane. What does it mean?

Answer 1

It most likely means that the vector points in the positive y direction.
That is that it is a scalar multiple of <0,0,1>.
Usually the 3d space is represented graphically (as in a 2d picture) with z pointing up, y pointing right and x pointing "out" of the paper.
Saying "right of the xz plane" is somewhat ambiguous, but it most likely means what I indicated.