operator and eigenvalue
Imagine a machine that takes in a shape and gives back a transformed shape: it might stretch it, rotate it, or flip it. Most shapes come out looking different. But a few special shapes come out pointing the same way as before, merely scaled up or down by some number. Quantum mechanics is built on exactly this picture. An operator is the machine that represents a measurable quantity, and the special states it leaves pointing the same way are the ones with definite values.
More precisely, an operator is a mathematical instruction that acts on a wavefunction — there is one for energy, one for momentum, one for position, and so on. When the operator acts on certain special wavefunctions, called eigenfunctions, it gives back the same wavefunction multiplied by a plain number. That number is the eigenvalue, and it is exactly the value you would measure for that quantity. The Schrödinger equation itself is one such relationship: the energy operator acting on a wavefunction returns that wavefunction times its energy.
The honest point is that this framework is what makes quantization fall out automatically: only certain eigenvalues are allowed, so only certain energies or momenta can be measured. If a system is not in an eigenstate of some quantity, then that quantity does not have a single definite value — a measurement will yield one of the allowed eigenvalues with a probability set by the wavefunction. This is the precise machinery behind the fuzzy probabilistic picture of the quantum world.
The hydrogen 1s orbital is an eigenfunction of the energy operator. Apply that operator to it and you get the same orbital back, multiplied by −13.6 electron-volts — its eigenvalue. That is why every hydrogen atom in the 1s state has exactly this energy: the orbital is a state with a single, sharply defined energy.
An eigenfunction comes back unchanged in shape, scaled by its measurable value.
Operators for measurable quantities are of a special kind (Hermitian) that guarantees their eigenvalues are real numbers — which makes sense, since any measurement gives a real result. Not all operators commute, and that fact is the deep reason behind the uncertainty principle.