Conceptual understanding of Schrödinger equation So I followed this lecture:
https://www.youtube.com/watch?v=qu-jyrwW6hw
which starts of with the statement:
If you have a Schrödinger equation for an energy eigenstate you have 
$$-\frac{\hbar}{2m}\frac{d^2}{dx^2}\psi(x) + V(x)\psi(x) = E \psi(x)\tag{1}$$
Question 1: What does it mean to have a energy eigenstate in this context? All eigenstates I ever cared about were the eigenstates and eigenfunctions of Hamiltonians.
Question 2: Is equation (1) a general statement or specific to some conditions? Usually I assumeed that Schrödingers-equation is used for time-evolutions but this doesn't seem to be the case here.
 A: This is the time independent Schrödinger equation. It is basically an eigenvalue problem
$$\hat{H}\psi=E\psi $$
where $$\hat{H}=-\frac{\hbar^2}{2m}\nabla^2+V(x)$$
is the Hamiltonian of the system. Since you yourself mentioned eigenstates of the Hamiltonian I'm going to guess you already know about why the Hamiltonian has this form. The solution of this equation are the eigenstates of the Hamiltonian operator, a set of eigenvectors and eigenvalues.
Probably, the Schrödinger equation you have in mind is the time dependent Schrödinger equation
$$i\hbar\frac{\partial}{\partial t}|\psi\rangle=\hat{H}|\psi\rangle $$ 
Why do we need two separate equations? Well, the true equation of motion of the state is the time dependent version, nevertheless, considering the appearance of the hamiltonian, it's useful to solve the time independent one and get the eigenstates. Why? Because once you have some states $|n\rangle$ such that $\hat{H}|n\rangle=E_n|n\rangle$ you can verify that
$$ |\psi(t)\rangle=\sum_n a_n\exp \left(-i\frac{E_n t}{\hbar} \right)|n\rangle$$
is a solution to the time dependent version for some coefficients $a_n$.
A: You have stated that the only eigenstates you care about are for the Hamiltonian.  That equation IS the Hamiltonian for a non-relativistic particle.  
The differential operator on the left hand side is H, the Hamiltonian operator, the E on the right hand side is the energy, a scalar number.  Solving this equation for the allowed wavefunctions and energies provides you with a complete set of eigenfunctions and eigenvalues {psi_n, E_n}.  By the way I think you are missing a square on your h_bar.
As for the time-dependent equation, its solutions can be built up from the eigenfunctions since the span Hilbert space.  This is a common approach to solving time dependent PDE, and is used in acoustics, optics and all other wave mechanics.
As for it being a special case? The only things special about the equation you have posted are (1) it is non-relativistic, (2) it is 1-dim (in 3-dim the second derivative would be replaced with the Laplacian operator), and (3) no boundary conditions are explicitly mentioned, e.g. psi(x0) = 0 or psi(infinity) = 0, etc.
