Quantum systems, like classical system, are described by "states". States contain information that allows us, given external forces, predict the state in any future time given the state now.
For classical particle, according to Newton laws, states are characterized by positions and velocities or, which is almost the same, positions and momenta. For a quantum particle they are characterized by a "wave functions" or "state vectors in a Hilbert space". (Newton's equations of motion are replaced by Schrodinger's equation.)
Thus for a classical particle the dimension of the state space is finite (for instance 3+3, 3 position components and 3 momemtum components), while for a quantum particle state space is infinite dimensional (because wave functions can have infinitely many independent "shapes").
For each state of a classical particle you can calculate energy that is associated with this state. You get a number. For quantum states you get, in general, a whole distribution of different energies. When this distribution concentrates on one particular value (like for a classical particle) - you have "energy state". The value of energy associated with such a state is called "energy level".