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][1]. 

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. 

Now suppose $I$ is a right ideal. Hence $i(I)$ is a left ideal and by the above, it's also a right ideal. Consequently, $I=i(i(I))$ is a left ideal and we are done.  






 

  [1]: http://www.ams.org/journals/proc/1979-076-02/S0002-9939-1979-0537074-2/S0002-9939-1979-0537074-2.pdf