Energy Eigenstate of a Hamiltonian

Yes. In linear algebra, operators (like the matrix you wrote down) act on vectors. For some matrix $M$ and vector $vec v$, if $Mvec v = lambda vec v$ for some scalar $lambda$, then we call $vec v$ and eigenvector of $M$ and $lambda$ its associated eigenvalue.

In quantum mechanics, the state of whatever system you're modeling is (roughly) represented as a vector in some vector space. In this context, it is common to refer to vectors as states and eigenvectors as eigenstates.