Confusion regarding plane of the paper I find terms like into the plane of paper out of the plane of paper How am I supposed to understand that?
 A: A good example of where this kind of terminology is often used for electrical current go into or out of the page (or, as you wrote, into or out of the plane of the paper), as shown in the diagram below:

Image source: Evan's Space Wonders of Physics
(disclosure: not my website)
In summary - anything coming out of the page is heading out of the page perpendicular to the flat page (or the plane of the paper), the opposite is true for going into the page.


A: The paper can be approximated as a two-dimensional object (or a plane). So the surface of the paper represents the plane. The area of that plane is a vector that points out normal to the plane surface. 
Now, any vector that points into the paper will be in a direction anti parallel to the area vector of the plane of the paper. Anything that points out of the paper will be in a direction parallel to the plane of paper. This convention holds for any of the one surface at a time. That surface is given by the plane you can see , so that the area vector points towards you.    
So, whenever there is something written as "points into the page" just look down to the plane of the paper. That's the direction of the vector. If it says "points out of the paper", simply look straight up from the paper surface. In both cases, you should consider only one surface at a time.
A: If you have a kleenex (or other rectangular) box, notice the line segments that connect the corners of the box all lie on the x-, y-, and z- axes (length, width, height or depth). So if you consider a slice of this volume, you will have a 2-d cross-section of the kleenex box; this can be the xy-plane. The z-axis comes into and out of this xy-plane. (Remember that x, y, z are all orthogonal to each other.) The xy-plane is your paper and the z-axis is into and out of your paper. 
