Schrodinger Equation

Time-independent

Describes stationary states and energy levels of a quantum system where the potential energy does not change over time
$\hat{H}\psi (x)=E\psi (x)$

1D

$-\frac{{\hslash }^2}{2m}\frac{d^2\psi (x)}{d{x}^2}+V(x)\psi (x)=E\psi (x)$

3D

$-\frac{{\hslash }^2}{2m}{\nabla }^2\psi (\mathbf{r})+V(\mathbf{r})\psi (\mathbf{r})=E\psi (\mathbf{r})$

Time-dependent

Describes how the quantum state of a physical system changes over time. Unlike the time-independent version, it can handle situations where the potential energy changes with time, or track the evolution of non-stationary states (such as a moving wave packet).
$i\hslash \frac{\partial }{\partial t}|\Psi (t)⟩=\hat{H}|\Psi (t)⟩$