Let wave function $\Psi$ be defined on domain $D \in \mathbb{R}^n$. The Neumann condition $\frac{\partial \Psi} {\partial {\bf n}} = 0$ on the boundary $\partial D$ has a simple interpretation in terms of the probability current of $\Psi$. For $\Delta \Psi = i \partial\Psi/\partial t$ (although it's usually taken as $i \partial\Psi/\partial t = - \Delta \Psi$), the probability current at an arbitrary point ${\bf x} \in \mathbb{R}^n$ is 
$$
{\bf j}({\bf x}) = i [ \Psi^*({\bf x}){\bf \nabla}\Psi({\bf x}) - \Psi({\bf x}){\bf \nabla}\Psi^*({\bf x}) ]
$$
and the normal current on $\partial D$ reads 
$$
{\bf n} \cdot {\bf j} = i\; [ \Psi^* \frac{\partial \Psi}{\partial {\bf n}} - \Psi \frac{\partial \Psi^*}{\partial {\bf n}} ]
$$
(has the wrong sign, I know, but I accounted for OP's form of the Sch.eq. as $\Delta \Psi = i \partial\Psi/\partial t$).

Setting $\frac{\partial \Psi} {\partial {\bf n}} = 0$ amounts to ${\bf n} \cdot {\bf j} = 0$ everywhere on $\partial D$, thus confining the corresponding system within $D$ without an infinite potential well, as under Dirichlet conditions ($\Psi = 0$ on $\partial D$). This is the case of perfect reflection on $\partial D$. 

There is a mention of this in Section 5.2 of *Visual Quantum Mechanics: 
Selected Topics with Computer-Generated Animations of Quantum-Mechanical Phenomena*, by Bernd Thaller ([Springer](http://www.springer.com/us/book/9780387989297), 2000); [Google Books link](https://books.google.com/books?id=GOfjBwAAQBAJ&pg=PR1&dq=Visual+Quantum+Mechanics:+Selected+Topics+with+Computer-Generated&hl=en&sa=X&ved=0CB4Q6AEwAGoVChMIlv7-5KOkxwIVxTY-Ch2jCgmI#v=onepage&q=Visual%20Quantum%20Mechanics%3A%20Selected%20Topics%20with%20Computer-Generated&f=false). 
 
As for applications, one answer to another post, http://physics.stackexchange.com/questions/30374/can-we-impose-a-boundary-condition-on-the-derivative-of-the-wavefunction-through?rq=1, pointed to the use of Neumann conditions in R-matrix scattering theory.