A standard commutative diagram is the pictorial representation of the objects and morphisms within a given category. (This is the most general definition I can provide.)
A specific commutative diagram can be found by looking at morphisms which are group isomorphisms in the category $\textbf{Grp}$. Other examples exist and can be found in the textbook Category Theory for the Working Mathematician.
My question is this: can we define, in a rigorous manner, a noncommutative diagram? And, if we can, should we? What would such a concept be?
Note: I think this is worth considering because the breakdown of commutativity is always interesting to see in algebra.
Update: The user going by the name Daniel T. has addressed this question properly. However, Randall's answer in the comments also pinpoints that the notion of a noncommutative diagram is essentially silly and asinine in the notion of category theory. As such, I apologize for wasting time.
A (not necessarily commutative) diagram is a functor $F:I\to \mathcal C$ where $I$ as a category with the shape you want (a triangle, a square, a rhombicosidodecahedron). Given morphisms $p:a\to b$, $q:b\to c$ and $r:a\to c$ we must have $F(g\circ f) = F(q)\circ F(p)$, but maybe $F(g\circ f) \neq F(h)$.
Definition: A diagram is commutative if for all chains $$ x\xrightarrow{f_0} a_0\xrightarrow{} \cdots \xrightarrow{f_n} a_n \xrightarrow{f_{n+1}} y $$ and $$ x\xrightarrow{g_0} b_0\xrightarrow{} \cdots \xrightarrow{g_m} b_n \xrightarrow{g_{m+1}} y $$ in $\mathcal C$ then $g_{m+1}\circ \cdots g_0 = f_{n+1}\circ\cdots f_0$. Otherwise it's a non-commutative diagram.