Take a linear map from some space to itself, and ask:
What lines (through the origin) are mapped back to themselves? Those are the eigenvectors, and the amount by which they're elongated or shortened are the eigenvalues.
So, if we talk about 3d space, and we rotate things - the rotation axis is unchanged. That's an eigenvector (with eigenvalue 1).
If we mirror things - any vector in the mirror plane remains unchanged, that's an eigenvector (with eigenvalue 1), the vector perpendicular to the mirror is unchanged, but flipped, so that's an eigenvector (with eigenvalue -1).
If we dilate everything along the x axis by a factor of 2, say, then the x axis is an eigenvector (with eigenvalue 2), while the y and z axis and any vector in that plane is an eigenvector (with eigenvalue 1). Any other vector is "tilted", so not mapped to itself, so not an eigenvector.
What lines (through the origin) are mapped back to themselves? Those are the eigenvectors, and the amount by which they're elongated or shortened are the eigenvalues.
So, if we talk about 3d space, and we rotate things - the rotation axis is unchanged. That's an eigenvector (with eigenvalue 1).
If we mirror things - any vector in the mirror plane remains unchanged, that's an eigenvector (with eigenvalue 1), the vector perpendicular to the mirror is unchanged, but flipped, so that's an eigenvector (with eigenvalue -1).
If we dilate everything along the x axis by a factor of 2, say, then the x axis is an eigenvector (with eigenvalue 2), while the y and z axis and any vector in that plane is an eigenvector (with eigenvalue 1). Any other vector is "tilted", so not mapped to itself, so not an eigenvector.