The following definition was essentially stated on the website mentioned above. This is what I have used to explain Von Mises to engineers and some nonengineers. I'll use engineering terms in parentheses. This is not rigorous, but it's accurate.
"Materials are generally tested by stretching a piece of metal (a uniaxial tension test). Real parts have much more complicated loads--twisting, bending, shearing, etc. Von Mises stress is a calculation which takes these complicated loads and figures out what their stretching (uniaxial tension) equivalent is. With this Von Mises stress, people can compare complicated loading (stresses) in a part to the simple values (uniaxial tension) which are published in books)."
Brad