String diagrams are used in two rather different contexts (that I know of).
Firstly, they are used for monoidal categories, where they typically look something like circuit diagrams and the important things are the boxes and the wires that connect them, and equations look like this:
Secondly, they are used for 2-categories (and for the category $\mathbf{Cat}$ in particular). Here the wires are functors, the boxes are natural transformations, and in addition the areas in between the wires are important, since they represent categories. An example of an equation is (from Marsden, 2014):
There are some small differences between these two types of notation, but in general they are very similar. In both cases the idea is that you can manipulate the diagrams according to rules that depend on exactly which categories you're dealing with, but which generally correspond to topologically intuitive operations like bending wires around each other, sliding boxes along the wires, and so on.
In my own informal notes, I also use string diagrams for categories and functors in a different way, where lines are categories and boxes are functors, and parallel lines represent product categories. I don't have any theorem proving that this results in sound proofs, but it seems to work quite well.
My question is simply, what is the relationship between these things? What do monoidal categories and 2-categories have in common that allows such similar notation to be used for both, and are there other cases, besides the ones mentioned here, where string diagrams can also be used?