An eigenvector of a matrix is a vector v that satisfies Av = Lv. In other words, after A "acts on" the vector v, the result is the vector v just scaled by the constant number L.
In this example, we find the eigenvectors of a given 3x3 matrix. This is done by finding the null space of the matrix A-LI. The null space solution (A-LI)x = 0 always results in an infinite number of solutions for the vector x. As such, we find a basis for each one of these solutions, and thus the "basis for an eigenspace" terminology.