What do zero eigenvalues mean?

Geometrically, having one or more eigenvalues of zero simply means the nullspace is nontrivial, so that the image is a "crushed" a bit, since it is of lower dimension.

Other than the obvious case of having exactly one 0 eigenvalue, there's no way to predict the dimension of the nullspace from the number of zero eigenvalues alone. The dimension of the nullspace is bounded by the multiplicity of zero eigenvalues, however.

For example, as mentioned in the comments, the matrices with all eigenvalues 0 are the nilpotent matrices, and these can have rank anywhere between $0$ and $n-1$