Power method to calculate eigenvectors
Whole books and fields of research are devoted to methods for computing eigenvalues and eigenvectors. Properties of the matrix matter, a lot. The ideal method will depend on whether


the matrices are symmetric or nonsymmetric, 
positive definite or indefinite, 
sparse or not sparse, 
you need all the eigenvalues/vectors or just a few, 
you have $O(1)$ access to the elements of the matrix or just the ability to apply it to vectors via a function call
etc...


That said, at the core of most modern methods for finding all the eigenvalues of a generic matrix is the The QR algorithm (the eigenvalue algorithm, not the related orthogonalization algorithm). The methods most used in practice such as the Arnoldi iteration (or Lanczos for symmetric problems) are basically a 2 step process in which the matrix is converted into a nicer form, and then the QR algorithm is used.

Understanding the power method is a good place to start from because it's intuition informs most of the other methods in some way or another.