The eigenvectors are the “axes” of the transformation represented by the matrix.

Consider spinning a globe (the universe of vectors): every location faces a new direction, except the poles.

An “eigenvector” is an input that doesn’t change direction when it’s run through the matrix (it points “along the axis”). And although the direction doesn’t change, the size might. The eigenvalue is the amount the eigenvector is scaled up or down when going through the matrix.

(Shameless plug, more here: http://betterexplained.com/articles/linear-algebra-guide/)