As I understand the term "derivation" in the context of Lie Algebra $L$ is a $\delta\in End(L)$ such that $\delta$ satisfies "Leibniz's rule" or what is often called product rule for differentiation.
The derivations form a Lie-subalgebra of $End(L)$ under the Lie-bracket.
There is also the derived subalgebra which is the Lie-subalgebra generated by $\{ [x,y]:x,y\in L\}$. It is somewhat analogous to the derived subgroup.
My question which I think is quite natural is what is the exact relationship between a derivation and derived subalgebra. Or to put it more clearly, since these names are very similar there must be some kind of relationship. (I thought that the set of derivations $Der(L)\subset End(L)$ is a derived subalgebra of $End(L)$. i.e. $Der(L)=[End(L),End(L)]$. But I am pretty sure, this is not quite true.
No, $Der(L)$ need not be $[End(L),End(L)]$. For an abelian Lie algebra $L$ of dimension $n$ over $K$ we have $Der(L)=End(L)$, but $[End(L),End(L)]=[M_n(K),M_n(K)]$, which is the vector space of traceless matrices of size $n$.
The derived ideal $[L,L]$ equals $ad(L)(L)$, where $ad(L)$ is the space of inner derivations.