Justin,

It is almost exactly like the "NaN" (Not a Number) value returned by standard "IEEE" floating point processors; it is a value returned when neither a numerical value or "infinity" makes any sense.
A nonmathematical analogy: it is like answering "I don't know" when somebody expected a person's name in response.

The only difference between Anderson's arithmetic and the IEEE version are in whether this special value is considered equal to itself; in Anderson's version it is, in the IEEE version it is not. (See the Wikipedia page on Anderson for details.) I find the IEEE approach more logical; if I ask "Who ate the last cookie?" and somebody says "I don't know" and if I later ask "Who discovered magnesium?" and get the same answer, I do not conclude that the discoverer of magnesium celebrated by eating the last cookie! Nonetheless, there may be applications in which Anderson's approach makes more sense.

Not that the term "nullity" means something completely different in linear algebra.

Good Hunting!
-RD