An example of a simple module which does not occur in the regular module? Let $K$ be a field and $A$ be a $K$-algebra.
I know, if $A$ is artinain algebra, then by Krull-Schmidt Theorem  $A$ , as a left regular module, can be written as a direct sum of indecomposable $A$-modules, that is 
$A=\oplus_{i=1}^n S_i$ where each $S_i$ is indecomposable $A$-module
Moreover, each $S_i$ contains only one maximal submodule, which is given by $J_i= J(A)S_i$, and every simple $A$-module is isomorphic to some $A/J_i$.
My question is that, can you,  please,  tell me an example of a non simisimple algebra, or a ring, such that it has a simple module which does not occur in the regular module.
By occur I mean it has to be isomorphic to a simple submodule of a regular module

If $S$ is a simple module over a ring $A$, and $x$ is a non-zero element of 
$S$, then the annihilator of $x$ is a maximal left ideal $\mathfrak m$ of $A$.  
Thus is $S$ embeds into $A$, then $A$ contains a non-zero element annihilated by $\mathfrak m$.
If $A$ is not a simple module over itself (i.e. is not a division algebra), then $\mathfrak m$ will contain non-zero elements, and so if $A$ is an integral domain (i.e. does not contain non-zero zero-divisors), then it will not contain any simple submodules.

The example $K[X]$ of a polyomial algebra that appears in the other answers is a special case of this general observation, since $K[X]$ is an integral domain.
