What is a simple lie algebra?
What should I be thinking of when I come across these? What is a good example or two that I should keep in the back of my mind at all times? I know they are useful, but I can't appreciate why, I don't understand the definition at all, don't know how to think about it. 'Simple means indivisible' is one step in the right direction.
In general algebra, an algebraic structure $A$ is said to be simple if every nonconstant homomorphism $A\to B$ is injective (i.e., has trivial kernel). E.g. a group is simple iff it has no nontrivial normal subgroup, a ring if it has no nontrivial ideal, etc. (these are the concepts that correspond to the kernel of some homomorphism).
For the case of Lie algebras, the Abelian simple ones (i.e. the 1 dimensionals) are excluded for technical reasons (see the link for the other question you referred).
An example of simple Lie algebra is $\Bbb R^3$ with vectorial product. Further examples can be found on wikipedia.