Short Definition
An eigenvalue is the scaling factor applied to an eigenvector by a linear transformation.
Intuition
Some directions keep pointing the same way after a transformation; the eigenvalue says how much that direction stretches or shrinks.
Technical Definition
For a matrix A and nonzero vector v, lambda is an eigenvalue if Av = lambda v.
Example
PCA uses eigenvalues to measure variance captured by principal directions.
Common Misunderstandings
Eigenvalues only make sense together with a transformation or matrix.
Large eigenvalues are not automatically better; interpretation depends on the problem.