Looks like it is easier to find example of commutative rings rather than
non-commutative rings. Prabably the easiest examples of the former are
$\mathbb{Z}$ and $\mathbb{Z}_n$. We can find elaborations on these two commutative rings in various
literatures including here and here. These are quite simple and easy to comprehend.
However, the examples on simple (non-commutative kind) are not that easy.
One example is found here and it has been mentioned as "one of the simplest examples of a non-commutative ring". This is also on the easier side.
EDIT: The following two are also commutative rings.
Other examples are given in this enumeration. But as you can see, examples like Gaussian integers or Eisenstein integers are difficult for starters to comprehend.
Do you think you can give one or two simple examples on non-commutative rings, based on every day numbers?
If it is that difficult, perhaps some insight comments why this is difficult would be welcome.
Best Answer
Given an abelian group $M$, let $\operatorname{End}(M)$ denote the set of all homomorphisms $M \to M$ (i.e endomorphisms). This set becomes a ring under pointwise addition and composition.
To see that $\operatorname{End}(M)$ may not be a commutative ring, choose another noncommutative ring $R$ (you already know one). Left multiplication by elements of $R$ are endomorphisms of the underlying abelian group. Since there are elements $a, b \in R$ such that $ab \ne ba$, $\operatorname{End}(R, +)$ is noncommutative.