**Generalized Eigenvector**

I'm surprised no one has yet mentioned the first example an undergraduate is likely to see.  Suppose $A$ is a linear map from a finite dimensional vector space $V$ to $V$, with eigenvalue $\lambda$.  Any nonzero vector in $\text{ker}(A-\lambda I)^k$ for some $k\ge1$ is called a generalized eigenvector for eigenvalue $\lambda$.  These are used in proving the existance of the Jordan Canonical Form.