Here's a slightly different example. If you are comfortable with doing algebra with polynomials, then I think you will find it easy to understand.
Take any finite group $G$ (say of order $n$). Now, you can formally make a set $\{\sum_{g\in G} \alpha_g g\mid \alpha_g\in \Bbb R\}$.
Since you know how to multiply the elements of $G$, you can extend the multiplication to this set by requiring linearity to hold. So, for example, if you have $a,b,c,d\in G$ and you want to multiply $(3a+2b)(c-5d)$, you would just distribute: $3ac-15ad+2bc-10bd$. Then $ac,ad,bc,bd$ would be multiplied to be elements of $G$, and you would wind back up in the set. Addition is just done by adding "like terms."
With these operations, the set is called the group ring $\Bbb R[G]$. In fact, $G$ doesn't have to be finite, but if $G$ is infinite then you need to only use finite sums in the set I gave above. Even more, $G$ doesn't have to be a group, it just needs to have an associatve multiplication defined on it, so it could be just a monoid or semigroup.
A monoid can informally be described by thinking of a group which does not require the existence of inverses. So, if $x$ is an indeterminate, then $S=\{1,x,x^2,x^3\dots\}$ is a monoid, because $x^ix^j=x^{i+j}$ is an associative multiplication. The monoid ring $\Bbb R[S]$ for this set is something you are really familiar with: it is usualy denoted $\Bbb R [x]$ and called "the ring of polynomials over $\Bbb R$"!
For another thing, you can use any ring you want besides $\Bbb R$! You could even use $\Bbb Z$ if you so chose, or whatever other ring you wanted.
I'm confident that if you stretch your intuition for multiplying polynomials by replacing the powers of $x$ with elements of a group (or monoid), you will quickly grasp what a group ring is.
Anyhow, let's get to the point. If you choose $G$ to be a group that is not Abelian (that is, there exists $g,h\in G$ such that $gh\neq hg$) then you know for sure $\Bbb R[G]$ is not abelian: because in the ring, $gh\neq hg$.
If you would like to experiment with the smallest group which isn't commutative, you'll have to begin with the symmetries of a triangle $S_3=\{1,\sigma,\sigma^2,\tau,\sigma\tau,\sigma^2\tau\}$. To review, the multiplication obeys the relations $\sigma^3=1=\tau^2$, and $\tau\sigma=\sigma^2\tau$.
Pick two elements $p,q$ of $\Bbb Z[S_3]$. Compute $\sigma p$ and $p\sigma$. Compute $p+q$ and $p-q$ and $pq$. Have fun!