In general, the condition number measures the spread of the Eigenvalues of a matrix. Matrices with condition numbers close to unity are well conditioned. Matrices with large condition numbers are ill-conditioned.
Most computers today use a 64-bit IEEE floating point number, which provides about 15 decimal digits of precision. If the Eigenvalues of the matrix differ by more than that amount, the solution will fail. As they approach that limit, precision is lost.
If your solution fails due to an ill-conditioned matrix, you may need to change how you are modeling the problem. Often this can be as simple as remeshing to avoid regions that differ significantly is size. In other cases, the values picked as unknowns may need to be changed.
One simple example is the cantilever beam. If you were to model this structure using solid elements whose equations represent displacements, you would have an ill-conditioned matrix. There would be a low Eigenvalue corresponding to the diving board behavior of the beam and a very high Eigenvalue corresponding to the speed of sound through the beam. By changing the model to beam elements, which use rotations as unknowns, the high Eigenvalues are eliminated.
NOTE: Because the condition number could potentially approach infinity, FMS computes it's inverse (which could approach 0).