Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License.
 adjective mathematics In which each
submodule is a directsummand .  adjective mathematics For which every
invariant subspace has an invariantcomplement , equivalent to theminimal polynomial beingsquarefree .  adjective mathematics Being a direct sum of
simple Lie algebras .  adjective mathematics Being a
linear algebraic group whoseradical of theidentity component istrivial .
Etymologies
from Wiktionary, Creative Commons Attribution/ShareAlike License
semi + simple
Examples

The action of a real semisimple lie group on a complex flag manifold, II: Unitary representations on partially holomorphic cohomology spaces by Joseph Albert Wolf

Consider the Lie Algebra of a semisimple algebraic group over a field of characteristic

"generalized SatoTate law," and that figuring out which generalized SatoTate law applies to a given family amounts essentially to computing a certain complex semisimple (not necessarily connected) algebraic group, the "geometric monodromy group" attached to that family.
hernesheir commented on the word semisimple
The definitions are a bit semicomplex. No. 2 is quite clear and simple, however.
October 23, 2011