potential-energy term
The potential-energy term is the part of the Hamiltonian that describes the forces a particle feels. It is written simply as V(x), a function that assigns an energy to each point in space: low where the particle is happy to sit, high where it is pushed away. The shape of this function — a well, a barrier, a steady slope — encodes the entire physical setting of the problem.
Unlike the kinetic term, the potential is usually a plain multiplication rather than a derivative: at each location you just multiply the wavefunction by the local value of V. This makes it the storyteller of the Hamiltonian. Want an atom? Use the attractive pull of a nucleus. Want a particle in a box? Use walls of infinite potential. Want a tunnelling problem? Stand up a barrier the particle classically cannot cross.
Forces in quantum mechanics enter only through this term, as the steepness of the potential. Where V drops sharply, the particle is pushed strongly; where V is flat, it feels no force and moves freely. By sculpting V you set the stage, and the Schrödinger equation then tells you the wavefunctions and energies that the stage allows.
Different potentials are different physics: a spring, an atom, a confining box.
Only differences in potential energy have physical meaning; you can add a constant to V everywhere and nothing measurable changes — it merely shifts every energy by the same amount. The zero point of the potential is a matter of convenience.