In the Schrödinger picture, the states evolve in time, and operators are constant; in the Heisenberg picture, states are constant, but operators evolve, the evolution in both cases depending on the Hamiltonian. In the interaction or Dirac picture both states and operators evolve, the operators according to the free part of the Hamiltonian and the states according to the part which represents the interactions (modulo some trickery needed because the two parts of the Hamiltonian don't, typically, commute). One may be a bit suspicious of nature's ability to make such nice distinctions, but the math seems to work.