Av = lambda v. The eigenvector's direction survives the transformation; the eigenvalue is the stretch.
Useful facts to have ready: the sum of eigenvalues is the trace, and their product is the determinant. Both give quick sanity checks and occasionally the whole answer.
Where it matters in finance. The eigendecomposition of a covariance matrix is principal component analysis: the largest eigenvalue is the dominant common factor, which for equities is usually "the market". Small eigenvalues indicate near-collinear directions, which is why estimated covariance matrices are badly conditioned.