A simple Lie algebra is a Lie algebra $\mathfrak{g}$ such that:
Equivalently, a simple Lie algebra is a simple object of LieAlg that also is nonabelian. Note that there are only two abelian Lie algebras whose only proper ideal is the zero ideal: the trivial Lie algebra (which is not a simple object in $Lie Alg$ either, since the zero ideal is not proper either) and the line (which is a simple object in $Lie Alg$ but is still not considered a simple Lie algebra).
Simple Lie algebras over an algebraically closed field of characteristic zero, like many other things in mathematics, may be classified by Dynkin diagrams. We have:
$\mathfrak{a}_n = \mathfrak{sl}_{n+1}$, the special linear Lie algebra? of rank $n$. We count this only for $n \geq 1$, since $\mathfrak{a}_0$ is the trivial Lie algebra (which is not simple but is still semisimple).
$\mathfrak{b}_n = \mathfrak{so}_{2n+1}$, the odd-dimensional special orthogonal Lie algebra of rank $n$. We count this only for $n \geq 2$, since $\mathfrak{b}_n = \mathfrak{a}_n$ for $n \lt 2$.
$\mathfrak{c}_n = \mathfrak{sp}_n$, the symplectic Lie algebra? of rank $n$. We count this only for $n \geq 3$, since $\mathfrak{c}_n = \mathfrak{b}_n$ for $n \lt 3$.
$\mathfrak{d}_n = \mathfrak{so}_{2n}$, the even-dimensional special orthogonal Lie algebra of rank $n$. We count this only for $n \geq 4$, since $\mathfrak{d}_n = \mathfrak{a}_n$ for $n \lt 2$ and $n = 3$, while $\mathfrak{d}_2 = \mathfrak{a}_2 \oplus \mathfrak{a}_2$ (which is not simple but is still semisimple).
$\mathfrak{e}_n$, an exceptional Lie algebra? that only exists for rank $n \lt 9$. We count this only for $n \geq 6$ (thus for $n = 6, 7, 8$ in all), since $\mathfrak{e}_5 = \mathfrak{d}_5$, $\mathfrak{e}_4 = \mathfrak{a}_4$, $\mathfrak{e}_n = \mathfrak{a}_{n-1} \oplus \mathfrak{a}_1$ (which is not simple but is still semisimple) for $2 \leq n \lt 4$, and $\mathfrak{e}_n = \mathfrak{a}_n$ for $n \lt 2$.
the exceptional Lie algebra?s $\mathfrak{f}_4$ and $\mathfrak{g}_2$, which exist only for those ranks.
If you want to classify simple objects in $Lie Alg$, then there is one other possibility: the line (which has no corresponding Dynkin diagram).
It is much more difficult to classify simple Lie algebras over non-closed fields, over fields with positive characteristic, and especially over non-fields.
A semisimple Lie algebra is a direct sum of simple Lie algebras. In particular, every simple Lie algebra is semisimple, but there are many more.
A Lie group is a simple Lie group if the Lie algebra corresponding to it under Lie integration is simple.