There is a standard book which contains everything about electrostatics, the Laplace/Poisson equation and boundary conditions: Classical Electrodynamics by J. D. Jackson. Get the book from the library of your choice, read all chapters labeled "Electrostatics", and you will find the answers to all your questions (if you are simulating this, you need to know all this stuff anyway).
The nature of the boundary condition depends on the system you are describing. It means something else if you are calculating heat flow than electrostatics - obviously!
Let's say you have a differential equation (e.g. the Poisson equation $\Delta\varphi(\vec r)=-\frac{\varrho(\vec r)}{\varepsilon_0}$ describing electrostatics, and you solve it for the function $\varphi(\vec r)$. At the boundaries of the region (e.g. a cylinder, a cube, etc.) you have to fix some property of $\varphi(\vec r)$.
- Neumann boundary condition: You fix $\frac{\partial\varphi(\vec r)}{\partial\vec n}=\text{const}$ along the boundary, where $\vec n$ is the normal vector to the surface. It's basically the derivative of $\varphi$ if you go straight away from the surface. It can be a different value for every $\vec r$.
- Dirichlet boundary condition: You fix $\varphi(\vec r)=\text{const}$. It can be a different value for every $\vec r$.
You can only fix one of those two, or the sum (this is called Robin boundary condition).
Physical examples for electrostatics:
- Neumann boundary condition: The aforementioned derivative is constant if there is a fixed amount of charge on a surface, i.e. $\frac{\partial\varphi(\vec r)}{\partial\vec n}=\sigma(\vec r)$.
- Dirichlet boundary condition: The electrostatic potential $\varphi(\vec r)$ is fixed if you have a capacitor plate which you connected to a voltage source. E.g. if you have two capacitor plates which are at 0V and 5V, respectively, you would set $\varphi(\vec r)=0$ at the first plate and $\varphi(\vec r)=5$ at the second plate. That way, you can calculate the capacitance.
For heat flow, fixing the field $u$ (=Dirichlet BC) means fixing the temperature. If you have elements in your system that have a fixed temperature, you would use that one.
If you lenses are based on electrostatics, they probably only have Dirichlet boundary conditions, because that's how you describe a capacitor plate. If you have outer boundaries which are not capacitor plates, you should use a Neumann BC = 0 (in this case, it has nothing to do with fixing the charge), because that's the best BC for simulating an "infinite" system.
Best Answer
The Neumann boundary condition is rarely relevant on drums or other membrane-based instruments, since it implies the membrane is not tightly fixed to the frame, which makes it very hard to produce a meaningful sound on that instrument. The same argument also goes for stringed instruments.
More commonly it's used to describe wind instruments, where it signifies that the end is open and that the mean displacement of air particles on that end during tone production is at a local maximum, i.e. $\partial_x \psi(0,t)=\partial_x \psi(L,t)=0$, as illustrated in the picture below (taken from here), which is exactly the Neumann boundary condition, as opposed to the Dirichlet condition which is equivalent to a closed pipe end.
This, of course, is only illustrated in the case of one dimension, but similar reasoning applies to the more general 3D case. Also, a similar picture holds for membrane instruments, but, as I said, they are mainly bound by the Dirichlet condition, so my answer focuses on wind instruments.