I've read that eigenfunctions are some kind of eigenvectors. OK. For example consider the wave equation:
$\frac{∂^2u}{∂t^2} = c^2 \frac{∂^2u}{∂x^2}$
We can write it in such a way that a linear operator equals zero:
$(\frac{∂^2}{∂t^2} - c^2 \frac{∂^2}{∂x^2})(u) = 0$
With this form, we don't have an equation like this: $Df=λf$
Where $D$ is an operaror, $f$ is a vector (a member of vector space of differentiable functions), and $λ$ is a scalar. This is the equation I expected to see.
So, where does the concept of eigenvalues and eigenfunctions arise from?
In an abstract vector space, as studied in linear algebra, an "eigenvalue", $\lambda$, for a linear transformation, A, is a number such that there exist a non-zero vector v, such that $Av= \lambda v$. Further the vectors, v, satisfying that equation are "eigenvectors of A corresponding to eigenvalue, $\lambda$.
Of course, we can add functions and multiply functions by numbers (scalars) so we can have vector spaces of functions- often called "function spaces"- and there are linear transformations on them, such as $\frac{df}{dx}$ or $\int_0^x f(t) dt$.
Those linear transformations can have eigenvalues and eigenvectors, which are called "eigenfunctions". For example, the linear transformation $\frac{d^2}{dx^2}$ has eigenvalue -1 and corresponding eigenfunctions, sin(x) and cos(x), because $\frac{d^2 sin(x)}{dX^2}= -sin(x)$ and $\frac{d^2 cos(x)}{dx^2}= -cos(x)$.