A ring $R$ is simple if it is it is a simple object in the category of $R$-$R$-bimodules.
This can be stated in more elementary terms in any of the following equivalent ways:
In constructive algebra, this is too strong; we must say:
A field, or a division ring, is simple.
If $D$ is a division ring, then the ring $M_n(D)$ of $n \times n$ matrices with entries in $D$ is a simple ring.
The Weyl algebra $k\langle x, y\rangle/(x y - y x - 1)$ over a field $k$ is a simple ring. (In different language: this is the ring of differential operators with polynomial coefficients in one variable $t$, obtained as the image of the ring homomorphism from the noncommutative polynomial ring $k \langle x, y \rangle$ to the ring of $k$-linear endomorphisms $Vect(k[t], k[t])$ that sends $x$ to the derivative operator $\frac{d}{d t}$ and $y$ to the multiplication operator $t \cdot -$.) An explanation of why this is simple may be found here at Qiaochu Yuan’s blog.