I found the coherence axioms for monoidal categories very confusing until I realized that what they are really trying to say is the following. Given a monoidal category $\mathcal{M}$ and a sequence $O_1, \dots, O_n$ of objects (ordered), there are various ways of making sense of $O_1 \otimes \dots \otimes O_n$ all of which correspond to a choice of parenthesization. If we have associator isomorphisms as in the definition of a monoidal category, then we can get functorial maps between one choice of parenthesization and another; however, one actually gets several maps corresponding to different choices of associators. So a priori the order of parenthesization matters only up to isomorphism, but the isomorphism is not uniquely determined a priori. The coherence axioms state that the isomorphism is unique (whichever choice of associators you make, you'll get the same maps); that is, $O_1 \otimes \dots \otimes O_n$ can be defined in a manner unique up to canonical isomorphism. In the symmetric monoidal case, the axiom states that $\prod_A O_a$ can be defined in a manner unique up to canonical isomorphism when $A$ is an arbitrary finite (unordered) set.
There is a more general notion than a commutative diagram in a higher category. For instance, in a 2-category (here, just a strict one), one can say that a square diagram is 2-commutative; this means that the two ways of going around a diagram are related by a natural transformation. This leads to the notion of a 2-fibered product. The notion of a 2-fibered product is important when you want to keep track of higher morphisms; for instance, it is the appropriate notion for stacks (which form a 2-category).
Another example is in topology; one can think of a 2-commutative diagram as a diagram which commutes up to a specified homotopy.
In a monoidal higher category, the coherence axioms become 2-commutative diagrams instead of plain commutative diagrams, and the 2-morphisms in the 2-commutativity themselves have to satisfy coherence conditions of their own. This is kind of messy, but Lurie has developed a way of hiding the coherence axioms of a monoidal category in DAG II (so as to generalize to $(\infty, 1)$-categories). I don't really understand enough to say too much here.