Eigenvectors are column vectors or unit vectors whose length/magnitude is equal to 1. They are also called right vectors. Eigenvalues are coefficients that are applied on eigenvectors which give these vectors different values for length or magnitude.
A matrix can be decomposed into Eigenvectors and Eigenvalues and this process is called Eigen decomposition. These are then eventually used in machine learning methods like PCA (Principal Component Analysis) for gathering valuable insights from the given matrix.