semisimple Lie algebra
If solvable algebras are the floppy, abelian-built ones, semisimple algebras are their rigid opposite — they contain no solvable rot at all. They are the cleanest, most symmetric Lie algebras, assembled out of indivisible simple pieces, and they are exactly the ones whose finite-dimensional representations behave as nicely as possible. The classical matrix algebras sl(n), so(n), sp(2n) are the prime examples.
A finite-dimensional Lie algebra L over a field of characteristic zero is semisimple if its radical (largest solvable ideal) is zero; equivalently, L has no nonzero abelian ideal. The decisive structure theorem says a semisimple L is a direct sum of simple ideals L = L_1 (+) ... (+) L_r, the sum being a Lie-algebra direct sum with [L_i, L_j] = 0 for i not equal to j. So semisimple just means a finite direct sum of simples.
Semisimplicity has a beautiful operational test (Cartan's criterion): L is semisimple if and only if its Killing form is nondegenerate. Over an algebraically closed field of characteristic zero, Weyl's theorem of complete reducibility holds — every finite-dimensional representation splits as a direct sum of irreducibles — and the simple summands are classified completely by Dynkin diagrams.
sl(2, C) is simple, hence semisimple; sl(2, C) (+) sl(2, C) is semisimple but not simple. By contrast gl(n, C) = sl(n, C) (+) (center) is not semisimple, because its 1-dimensional center is a nonzero abelian ideal.
Simple is a single block; semisimple is a direct sum of blocks; gl(n) carries extra abelian baggage.
Every semisimple Lie algebra equals its own derived algebra, [L, L] = L, which is the polar opposite of solvable (where iterated derived algebras die). Reductive algebras are direct sums of a semisimple part and an abelian center; gl(n) is reductive but not semisimple.