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.