In this publication I found an explanation of the Hessian matrix, along with what it means for it to be ill-conditioned. In the paper, there is this link given between the error surface and the eigenvalues of the Hessian matrix:
The curvature of the error surface is given by the eigenvalues $\lambda_i$ of the Hessian matrix.
so it gives me a bit of hint as to why it might be important to care if it is poorly conditioned. But I'm not quite there yet, I have troubles seeing the consequences of an ill-conditioned Hessian.
So my question is: could you give me some intuitive understanding why should we care? In particular, in what models and how it can cause problems?