Von Mises Stress is actually a misnomer. It refers to a theory called the "Von Mises - Hencky criterion for ductile failure".
In an elastic body that is subject to a system of loads in 3 dimensions, a complex 3 dimensional system of stresses is developed (as you might imagine). That is, at any point within the body there are stresses acting in different directions, and the direction and magnitude of stresses changes from point to point. The Von Mises criterion is a formula for calculating whether the stress combination at a given point will cause failure.
There are three "Principal Stresses" that can be calculated at any point, acting in the x, y, and z directions. (The x,y, and z directions are the "principal axes" for the point and their orientation changes from point to point, but that is a technical issue.)
Von Mises found that, even though none of the principal stresses exceeds the yield stress of the material, it is possible for yielding to result from the combination of stresses. The Von Mises criteria is a formula for combining these 3 stresses into an equivalent stress, which is then compared to the yield stress of the material. (The yield stress is a known property of the material, and is usually considered to be the failure stress.)
The equivalent stress is often called the "Von Mises Stress" as a shorthand description. It is not really a stress, but a number that is used as an index. If the "Von Mises Stress" exceeds the yield stress, then the material is considered to be at the failure condition.
The formula is actually pretty simple, if you want to know it: (S1-S2)^2 + (S2-S3)^2 + (S3-S1)^2 = 2Se^2 Where S1, S2 and S3 are the principal stresses and Se is the equivalent stress, or "Von Mises Stress". Finding the principal stresses at any point in the body is another part of the problem.
