The difference between quartersawn and plainsawn lumber
Quartersawn:
The term "quartersawn" originated from the miliing practice of initially cutting a squared tree bole through the pith at right angles, resulting in four "quarters". Each of these quarters was then turned and cut so that the wide side of the resulting board showed straight grain. The piece would be turned for each (or every few) subsequent cut(s).
Boards with straight grain on the wide side are desirable primarily because they are stronger, but also because they cup less when drying.
Plainsawn:
The term "plainsawn" refers to the milling practice of cutting the squared bole repeatedly without turning it. This is quicker (and therefore less expensive) than quartersawing and also yields wider boards. Boards cut from the middle of the bole show grain like quartersawn lumber. Boards cut from the sides have face grain on the wide side.
Boards with face grain on the wide side are desirable for drawer fronts, cabinets, and other furnishings where the appearance is more important than strength.
In common usage, the terms "quartersawn" and "plainsawn" are used to describe the grain orientation of an individual board, regardless of how it was actually milled. For square stock, such as pen and bowl blanks, there is no wide side, and therefore no difference whatsoever between quartersawn and plainsawn.
I hope that helps,
Eric