Name a ring of 2 by 2 matrices where $a^3 = a$ and a belonging to this ring?
There is a theorem due to Jacobson that says if for every $a\in R$ there exists an $n\in\mathbb{N}$ such that $a^n=a$, then $R$ is commutative. (See this, or this for example).
Obviously the identity matrix cubed is itself... is this the sort of thing you're looking for?!
In general matrix rings are going to have a lot of idempotent elements $e$ such that $e^2=e$, and for all of those $e^3=e$ as well.
For an example where $a^2\neq a$, you could use $\begin{bmatrix}0&1\\1&0\end{bmatrix}$.