the so called time-depedent Schrödinger equation describes the time-evolution of a quantum system. It is, in a way, the true Schrödinger equation :
$$i\hbar \frac{\text{d}}{\text dt} |\psi\rangle = \hat{H}(t)|\psi\rangle$$
for a system whose Hamiltonian $\hat{H}$ has no explicit time dependence, it makes sense to try and find stationary states, whose time evolution is given by $|\psi(t)\rangle = e^{-iEt/\hbar}|\psi_0\rangle$. Plugging this ansatz in the (time-dependent) Schrödinger equation, you get the so-called time-independent Schrödinger equation :
$$\hat{H}|\psi_0\rangle = E|\psi_0\rangle$$
This is an eigenvalue problem so solving for $E$ and $|\psi_0\rangle$ will give you the energies for which a stationary state exists, as well as the wavefunctions of the said states.
Lastly, for a system with time-independent Hamiltonian, finding the stationary states allows to formally solve the full Schrödinger equation. Indeed, since $\hat{H}$ is hermitian, the spectral theorem ensures that we can find an orthonormal basis $|n\rangle$ of stationary states with energies $E_n$. Then, any state $|\psi(0)\rangle$ can be expanded over this basis :
$$|\psi(0)\rangle = \sum_n a_n |n\rangle \qquad \text{with} \quad a_n = \langle n |\psi(0)\rangle$$
You can check that the solution of the Schrödinger equation is then :
$$|\psi(t)\rangle = \sum_n a_ne^{-iE_nt/\hbar}|n\rangle$$
Edit : additional explanations on time dependent Hamiltonians, to answer questions from the comments below.
If $\hat H(t)$ is a time dependent Hamiltonian, then the spectral theorem still applies : at each instant $t$, we can find an orthonormal basis eigenvectors $|n(t)\rangle$ of $\hat{H}(t)$ with eigenvalues $E_n(t)$. Since the state $|n(t)\rangle$ may have an arbitrary time dependence, they are not called stationary states.
Since $\{ |n(t)\rangle\}$ is a basis, we can expand $|\psi(t)\rangle$ on it at each time $t$ :
$$|\psi(t)\rangle = \sum_n a_n(t) |n(t)\rangle \qquad \text{with}\quad a_n = \langle n(t) |\psi(t)\rangle$$
The $a_n$ verify a set of differential equations. However, contrary to the time-independent case, those equations are coupled. Explicitely, by plugging the expansion of $|\psi(t)\rangle$ in the Schrödinger equation and take the product with $\langle n(t) \rangle$, we get :
$$\dot{a}_n= -\frac{iE_n}{\hbar}a_n - \sum_k a_k \bigg\langle n(t)\bigg|\frac{\text d}{\text d t}\bigg|k(t) \bigg\rangle $$
Some approximations can be made to simplify those equations and find some solutions (look up adiabatic theorem)
