An operation E on a space U is idempotent if repeating it produces no additional effect: E(E(x))=E(x) for all x's in E's domain, or, more succinctly (and especially for linear operators), E2=E. A square matrix M is a linear operator, so we call M idempotent if M2=M.
The only operator X which is both nilpotent and idempotent is 0: We have Xj=X for all j>0, because X is idempotent, but we also have Xk=0 for some k, because X is nilpotent.