snipez90's reply is on the mark. Dickfore, those aren't really layman's terms :/
The definition is pretty easy to use.
Suppose you have a linear transformation T. The nullspace of T is the set of vectors v where T(v) = 0.
For many common operations, the nullspace is {0} (transformations, shears, non-zero scalings, reflections). The nullspace can never be empty, because it must always contain the 0 vector (why? because at the least, T(0) = 0).
When the nullspace contains more than just {0}, it means multiple vectors map to zero. As a consequence of this, the transformation is not invertible. The reason is because if v /= u, T(u) = 0, and T(v) = 0, you don't know whether T^-1(0) equals u or v. In effect, the transformation throws away information about its argument.
The simplest example of a nullspace greater than {0} is a projection transformation. Think of a 2D painting of a 3D landscape. That is effectively a projection -- a kind of linear transformation -- that maps points from the scene to points on a canvas. However, when this is done, a dimension is effectively "lost" in the painting. You can't tell if a mountain is small and near to the artist or if it is large and far away.
If you took a ruler from the artist's eye to a point on the mountain, from the artist's point of view, the ruler would look like a single point (because he's viewing the end of the ruler, not the side). This ruler embodies the nullspace -- an entire dimension maps to a single point (the zero point).
In this example, the nullspace would be a 1 dimensional subspace. Notice the math there: a 3D scene becomes a 2D painting and the resulting nullspace is 1D. 3 = 2 + 1. We have names for these different values. The space is 3D. The "rank" of the transformation (our projection) is 2. The "nullity" of the transformation is 1. The dimension of the space is always the sum of the rank and nullity.