All of the above mean the same thing: a solution of the Schrödinger equation. The term state is often used when dealing with spectroscopic transitions, e.g. when an electron is excited from a state with a lower energy into one with a higher energy, i.e. it changes from a wave function with a smaller to one with a higher value of n. The term orbital (a word coined in reference to the distinct electron orbits of Bohr's model of the atom on which electrons were supposed to move without radiating) stresses the interpretation of the wave function in terms of the probability of locating an electron. This often occurs in the context of chemical bonds.
The energy of a state with quantum number n is . For transitions between two states, the energy difference between the two states must either be supplied (excitation) or emitted. Since the states have discrete energy levels, only fitting energy quanta can be involved in the transition. The energy difference between to adjacent states is , so that spectral lines occur only at the corresponding energies (frequencies, wave numbers,...).
This had been observed long before quantum mechanics - the first observations concern the Sun's spectrum. The constants in front of the bracket are collectively known as the Rydberg constant, RH; its value was determined experimentally before quantum physics provided the link to the fundamental constants.
The different series of spectral lines discovered by Lyman, Balmer, and Paschen, correspond to the excitations from the n=1,2,3 states.
Now let's have a look at the shapes of the electron probability clouds of the various states.