What is a broad idea of what a simple ring is?
I am looking mostly for something intuitive rather than a formal mathematical answer.
For example the broad intuition of what a coset is, is that it's a subgroup that has been offset by an element of the bigger group.
Are you familiar with simple groups? The idea behind a simple group is that its only normal subgroups are the entire group and the trivial group $\{1\}$. The idea behind a simple ring is much the same: the only two-sided ideals are $(0)$ and the entire ring.