nLab
braiding

Any braided monoidal category has a natural isomorphism

$$B_{x,y}:x\otimes y\to y\otimes x$$B_{x,y} : x \otimes y \to y \otimes x

called the braiding.

A braided monoidal category is symmetric if and only if $B_{x,y}$ and $B_{y,x}$ are inverses (although they are isomorphisms regardless).

For example, in Vect, the braiding maps $a\otimes b$ (a typical generator of $x\otimes y$) to $b\otimes a$. But a braiding is most interesting when it does not look like something trivial like that.