Most directions in space, when a linear transformation is applied to them, change: a rotation sends them somewhere else, a shear tilts them, a projection drops them onto a subspace. But for almost every linear transformation there are a handful of special directions that survive — directions the transformation merely scales without rotating or reflecting them. These are the eigenvectors, and the scaling factors are the eigenvalues. The German prefix eigen- (own, characteristic, proper) was attached by David Hilbert around 1904. Once you know a transformation's eigenvectors and eigenvalues, you know almost everything about it.
Formally, an eigenvector of a linear transformation T is a nonzero vector v such that T(v) = λv for some scalar λ — the corresponding eigenvalue. Geometrically, T merely stretches v by the factor λ (negative λ flips the direction; complex λ adds rotation in 2D real space). To find them, you solve the characteristic equation det(T − λI) = 0 — a polynomial whose roots are exactly the eigenvalues. Each eigenvalue corresponds to an eigenspace of vectors that all scale by the same factor. The spectral theorem — one of the most consequential results in linear algebra — says that a real symmetric matrix has a complete orthonormal basis of eigenvectors with real eigenvalues. Equivalently, every symmetric matrix can be diagonalized: written as PDP⁻¹ where D is a diagonal matrix of eigenvalues and P's columns are eigenvectors. In eigenvector coordinates, the transformation is diagonal — it acts independently on each axis, which is the simplest possible behavior. Generalizations include Jordan normal form (when full diagonalization fails), singular value decomposition (eigenvalues for non-square matrices, the workhorse of numerical linear algebra), and spectral decomposition in infinite-dimensional Hilbert spaces (the foundation of quantum mechanics). The deep observation is that eigenvalues are intrinsic: they don't depend on the basis you used to write the matrix, only on the transformation itself. They are what the transformation truly is, abstracted from coordinates.
Principal Component Analysis — the oldest and still most-used dimensionality-reduction technique — finds the directions of greatest variance in a dataset by computing the top eigenvectors of the covariance matrix. PageRank finds web pages of greatest importance as the dominant eigenvector of a hyperlink-graph matrix. Quantum mechanics identifies physical observables (position, momentum, energy, spin) with linear operators, and the eigenvalues of those operators are the values that measurements can return. Vibration modes of a structure (whether a building under earthquake load or a guitar string) are eigenvalues of a stiffness matrix. Markov chains converge to a stationary distribution given by the eigenvector with eigenvalue 1. Spectral graph theory studies networks through the eigenvalues of their adjacency or Laplacian matrices.