When is matrix multiplication commutative?

Two matrices that are simultaneously diagonalizable are always commutative.

Proof: Let $A$, $B$ be two such $n \times n$ matrices over a base field $\mathbb K$, $v_1, \ldots, v_n$ a basis of Eigenvectors for $A$. Since $A$ and $B$ are simultaneously diagonalizable, such a basis exists and is also a basis of Eigenvectors for $B$. Denote the corresponding Eigenvalues of $A$ by $\lambda_1,\ldots\lambda_n$ and those of $B$ by $\mu_1,\ldots,\mu_n$.

Then it is known that there is a matrix $T$ whose columns are $v_1,\ldots,v_n$ such that $T^{-1} A T =: D_A$ and $T^{-1} B T =: D_B$ are diagonal matrices. Since $D_A$ and $D_B$ trivially commute (explicit calculation shows this), we have $$AB = T D_A T^{-1} T D_B T^{-1} = T D_A D_B T^{-1} =T D_B D_A T^{-1}= T D_B T^{-1} T D_A T^{-1} = BA.$$

The only matrices that commute with all other matrices are the multiples of the identity.

Among the groups of orthogonal matrices $O(n,\mathbb R)$, only the case $n=0$ (the trivial group) and $n=1$ (the two element group) give commutative matrix groups. The group $O(2,\mathbb R)$ consists of plane rotations and reflections, of which the former form an index $2$ commutative subgroup, but reflections do not commute with rotations or among each other in general. The largest commutative subalgebras of square matrices are those which are diagonal on some fixed basis; these subalgebras only have dimension $n$, out of an available $n^2$, so commutation is really quite exceptional among $n\times n$ matrices (at least for $n\geq2$). Nothing very simple can be said that (non-tautologically) characterises all commuting pairs of matrices.

Added. In fact the statement above about the largest commutative subalgebra is false. If you take the set of matrices whose nonzero entries occur only in a block that touches the main diagonal (without containing any diagonal positions) then this is always a commutative subalgebra. And then you can still throw in multiples of the identity matrix. Thus there is for instance a commutative subalgebra of dimension $\lfloor\frac{n^2}4\rfloor+1$ inside $M_n(K)$, for every $n$, and $\lfloor\frac{n^2}4\rfloor+1>n$ for all $n>3$. See here.