The Schrödinger equation tells us how a quantum state changes with time. In one common form, it is written as $$i\hbar\frac{\partial \psi}{\partial t} = \hat{H}\psi$$ where $\hat{H}$ is the Hamiltonian, representing the total energy.

If you know the initial wavefunction and the Hamiltonian, you can calculate how the system evolves. This makes the Schrödinger equation the central rule for nonrelativistic quantum mechanics.

The Hamiltonian is an operator that usually includes kinetic energy and potential energy. It plays a similar role to the energy function in classical physics, but it acts on wavefunctions.