Definitions
from Wiktionary, Creative Commons Attribution/ShareAlike License
 adj. In which each submodule is a direct summand.
 adj. For which every invariant subspace has an invariant complement, equivalent to the minimal polynomial being squarefree.
 adj. Being a direct sum of simple Lie algebras.
 adj. Being a linear algebraic group whose radical of the identity component is trivial.
Etymologies
semi + simple (Wiktionary)
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