A good place to start understanding these concepts on a more intuitive level is this video series, but I'll give a brief outline here of what the concepts mean.
Eigenvectors of a matrix are vectors which, when transformed by the matrix, are scaled by a constant. Eigenvalues are the constants by which they are scaled.
So if I have a matrix that rotates a vector $30^\circ$ around the x-axis, its only eigenvector is $\langle1, 0, 0\rangle$, and the corresponding eigenvalue is $1$.
You can find tons of explanations of how to actually calculate the eigenthings of a matrix just by some googling, so I'll leave that to you.
Does this explanation help?
Also, for the matrix you gave specifically, it has eigenvectors $\langle3, 2\rangle$ and $\langle1, 1\rangle$ with eigenvalues $2$ and $1$ respectively. This means that any scalar multiple of $\langle3, 2\rangle$ will be doubled when transformed by the matrix, and any scalar multiple of $\langle1, 1\rangle$ will be unchanged.