Every linear algebra class
— David Robinson (@drob) March 28, 2016
Me: What are eigenvectors
Teacher: You can think of them as an ndimensional kernel subspace
Me: No I can't
(Thanks to a discussion with a couple of people with locked Twitter accounts who inspired this post. The widget below might not be an original idea, but I checked at least three websites on Google about eigenvectors before writing it.)
Some people are happy with learning that, given some matrix A, a nonzero vector v is an eigenvector of A if Av = λv for some constant λ, and that constant is called the eigenvalue.
That is too abstract for a lot of people. Some might get the concept by thinking of the matrix as a transformation. A matrix transformation applied to a vector might stretch it, shear it, rotate it, etc. But if the effect of a matrix transformation on a particular vector is simply that it scales its size, then that vector is an eigenvector of the matrix, and the eigenvalue is the factor by which it gets scaled.
That explanation still leaves a lot of people wondering, "But what is an eigenvector?" And if you have some matrix in front of you, and you want to see what the eigenvectors are, well, set the determinant of A minus lambda I equal to zero....
The widget below is my attempt to make it possible to "see" eigenvectors (when they're real). The idea is to show a 2 × 2 matrix transformation applied to the whole plane. The direction of a black arrow represents the transformation of the vector (x, y).
This is a somewhat unsatisfactory representation, because the transformation doesn't happen "in place" as shown. But it means that you can visually pick real eigenvectors, by looking for a direction in which the black arrows line up with the identitymatrix grey arrows (or when they point in opposite directions – an eigenvector can be multiplied by a scalar and remain an eigenvector, and multiplying by 1 will make it point in the opposite direction). The default setting is the identity matrix, where the two sets of arrows perfectly overlap.
There might not be any direction in which the vectors are simply scaled by the matrix – a rotation is an example of this. In such cases, eigenvectors and eigenvalues will have imaginary components, and appear as complex conjugate pairs. I don't (yet?) have a good visual explanation of these that doesn't involve calculus.
You can enter (real) numbers into the textboxes representing the matrix below, or click on one of the preset options.
Identity  Rotate 90°  Reflect x  Shear

Update 
Currently showing [1, 0; 0, 1]. Show eigenvectors.
Eigenvector(s):  
Posted 20161224.