Psy-Kosh, Stephen: A finite-dimensional complex matrix has a complete basis of eigenvectors (i.e. it is diagonalizable) if and only if every generalized eigenvector is also an eigenvector. Intuitively, this means roughly that there are n independent directions (where n is the size of the matrix) such that vectors along these directions are stretched or shrunk uniformly by the matrix.

Try googling "jordan normal form", that may help clarify the situation.

I don't know the answer in the infinite-dimensional case.