A standard example of a non-commutative operation is matrix multiplication. Consider the following two integer matrices: $$ A=\begin{pmatrix} 1 & 1\\ 0&1 \end{pmatrix},\qquad B=\begin{pmatrix} 0 & 1\\ 0 & 1 \end{pmatrix} $$

If we compute $AB$ we get $$ AB=\begin{pmatrix} 0 & 2 \\ 0 & 1 \end{pmatrix} $$ but if we compute $BA$ we have $$ BA=\begin{pmatrix} 0 & 1 \\ 0 & 1 \end{pmatrix}. $$

Since $AB\neq BA$ we conclude that matrix product is not commutative.

Operations do not necessarily have to operate on numbers. Another classic example is function composition. Let $f$ and $g$ be real functions given by $$ f(x) = x^2,\qquad g(x) = 2x. $$

We see that $$ (f\circ g)(x) = f(g(x)) = (2x)^2 = 4x^2, $$ but $$ (g \circ f )(x) = g(f(x)) = 2(x^2) = 2x^2. $$ Since we got different functions, we conclude that function composition is not commutative.