There are many classes of algebras for which derivations have been studied.
Generally, if $A$ is an algebra and $a\in A$, the map $\delta_a:x\in A\mapsto ax-xa\in A$ is a derivation, which we call the inner derivation corresponding to $a$, and a somewhat boring one — for its existence does not tell us anything about the algebra. So usually what we do is consider the vector space $\def\Der{\operatorname{Der}}\Der(A)$ of all derivations, and its subspace $\def\InnDer{\operatorname{InnDer}}\InnDer(A)=\{\delta_a:a\in A\}$ and consider the quotient $\Der(A)/\InnDer(A)$ which, in a sense, tells us how many interesting derivations there are. We call $\Der(A)/\InnDer(AA)$ the vector space of outer derivations (even though its elements are not really derivations but classes of derivations) and write it $\def\OutDer{\operatorname{OutDer}}\OutDer(A)$.
The vector space $\OutDer(A)$ is an important invariant of $A$, and in fact it coincides with the so called first Hochschild cohomology group $HH^1(A)$ of $A$, which shows up all over the place.
If you want examples:
If $A=M_n(k)$ is the algebra of $n\times n$ matrices over the ground field, then $HH^1(A)=0$. This means, precisey, that every derivation of $A$ is an inner derivation.
If $L$ is a field extension of $k$ which is separable, then $HH^1(L)=0$ again. If the extension is nott separable, then the resul depends very much on what extension it is exactly.
If $A$ is the Weyl algebra, that is, the algebra of differential operators (on the line, say) witth polynomial coefficients, then $HH^1(A)=0$. This is a famous result of Dixmier.
If $A$ is an exterior algebra $\Lambda V$ on a vector space $V$, then on can easily compute $HH^1(A)$ but I do not recall the result.
And so on and on.
(My examples maybe make it seem like $HH^1$ is always zero, but it is certainly not!)
