String diagrams are a graphical calculus for expressing operations in a monoidal category. The idea is roughly to think of objects in a monoidal category as “strings” and a morphism from one tensor product to another as a node which the source strings enter and the target strings exit. Further structure on the monoidal category is encoded in geometrical properties on these strings. For instance
putting strings next to each other denotes the monoidal product, and having no string at all denotes the unit;
braiding strings over each other corresponds to – yes, the braiding (if any);
bending strings around corresponds to dualities on dualizable objects (if any).
There are many additional structures on monoidal categories (see the article by Selinger below for an overview) which encode further geometric properties. For instance
in monoidal categories which are ribbon categories the strings from above behave as if they have a small transversal extension which makes them behave as ribbons. Accordingly, there is a twist operation in the axioms of a ribbon category and graphically it corresponds to twisting the ribbons by 180 degrees.
in monoidal categories which are spherical all strings behave as if drawn on a sphere.
etc.
Many operations in monoidal categories that look rather unenlightening in symbols become very obvious in string diagram calculus, such as the trace: an output wire gets bent around and connects to an input.
Since a monoidal category is just a special case (namely the one-object case) of a bicategory, there is also a string diagram calculus for bicategories. This makes it manifest that the string diagram notation is Poincaré dual to the globular notation: where one uses -dimensional symbolds the other uses -dimensional symbols.
The Catsters (Simon Willerton), String diagrams (YouTube)