The directions a transformation leaves pointing the same way, and how much it stretches them. It sounds abstract and it has a concrete use: finding the directions in which your data varies most, which is what dimensionality reduction is built on.
Technical view
For a square matrix, vectors whose direction is unchanged under the transformation, scaled by the corresponding eigenvalue. Principal component analysis is the eigendecomposition of the covariance matrix, with components ordered by eigenvalue. Related quantities appear in analysing training stability, where the largest eigenvalue of the loss Hessian bounds the learning rate that can be used without divergence.
Related
Part of AI Terminology Index, section: Maths and Statistics.