energy level
Think of standing on a staircase rather than a ramp. On a ramp you can stop at any height you like; on a staircase you can only stand on a step — never halfway between two steps. An electron in an atom is on a staircase, not a ramp: it can only have certain fixed energies, called energy levels, and nothing in between is allowed.
An energy level is one of the allowed, discrete amounts of energy an electron may have within an atom. The lowest level is the ground state; higher ones are excited states. An electron can move from one level to another only by absorbing or emitting exactly the energy difference between them, no more and no less — which is why energy is exchanged in fixed-size packets.
This quantisation is the foundation of atomic spectra, chemical bonding, and why atoms are stable rather than spiralling into the nucleus. The caveat is that the steps are not evenly spaced: levels crowd together as you climb, and at the very top they merge into a continuum where the electron is no longer bound to the atom at all.
In a hydrogen atom the ground level sits at −13.6 electronvolts, the next at −3.4, then −1.5, and so on, crowding ever closer toward zero. An electron resting at −3.4 can drop to −13.6 only by releasing a photon carrying exactly the 10.2 eV difference — which appears as one precise line in hydrogen's spectrum.
Electrons stand on steps, not a ramp — only fixed energies are allowed.
The negative numbers simply mean the electron is bound: zero energy is set at the point where it just escapes the atom, so any bound state lies below zero.