I would like to know if when we take a second partial derivative of a function do we always get:
$$\frac{\partial^2{u}}{\partial{x} \, \partial{y}} = \frac{\partial{^2u}}{\partial{y} \, \partial x}$$
if not, what does it mean about the function if the condition happens or not?
It is true if the function is of class $C^2$.
It is called Schwarz theorem.
A counter-example if the function is not $C^2$ would be:
$$f\colon(x,y)\mapsto\begin{cases} xy\frac{x^2-y^2}{x^2+y^2} &\text{ if $(x,y)\ne (0,0)$} \\ 0 &\text{ otherwise.} \end{cases}$$
For info, the graph of $f$ looks like this:
