A "natural" example is given by the group ring $\mathbb{F}_2[Q]$ of the Quaternion group of order 8.
For, we have to show that each left ideal is also a right ideal, and conversely, each right ideal is also a left ideal. The first half (i.e. left is right) is shown in this paper.
Let $\mathfrak{L}$ resp. $\mathfrak{R}$ be the set of left resp. right ideals of $\mathbb{F}_2[Q]$. By the above, we have an injective map $\mathfrak{L} \to \mathfrak{R},\;I \mapsto I$ between finite sets. Hence it suffices to show that $\mathfrak{L}, \mathfrak{R}$ have the same cardinality.
Let $i:\mathbb{F}_2[Q] \to \mathbb{F}_2[Q],\;g \mapsto g^{-1}$ be the antipode. It's a general fact that for a left (right) ideal $I$, $i(I)$ is a right (left) ideal. Hence $i$ induces maps $f: \mathfrak{L} \to \mathfrak{R}, I \to i(I)$, $g: \mathfrak{R} \to \mathfrak{L}, I \to i(I)$ which are inverse to each other since $i^2 = \text{id}$.