Observable: a fancy word for "thing you can measure"
An observable is simply a physical quantity you could, in principle, point a real instrument at and get a reading: position, momentum, energy, the direction a tiny magnet points. The previous guide gave you the key idea — each such quantity is represented by an operator, a verb that acts on the state. "Observable" is just the word we use when we want to stress the *physics* (it is measurable) rather than the *math* (it is an operator). Same object, two hats.
The deep and surprising question is: *which* numbers can a measurement actually produce? In classical physics this is dull — an energy could be any value at all, smoothly. In quantum physics it is the whole drama. For many observables only a fixed, often spread-out menu of values is allowed. Energy of an atom comes in discrete steps; the spin of an electron, measured along any line, comes out only "up" or "down," never anything in between. The job of this guide is to say where that menu comes from.
Eigenstates: the states with a definite answer
Recall the magical states from last time — the ones an operator hands back unchanged in shape, merely scaled by a number. These deserve a name. A state that an observable's operator returns *as itself, times a constant* is called an eigenstate of that observable. The German prefix "eigen-" means "own" or "characteristic": an eigenstate is a state that is *characteristically its own* under that operator — the operator does not bend it into a new shape, it only rescales it.
And that scaling constant — the number the state gets multiplied by — is the eigenvalue. Here is the punchline, the single most important sentence in this guide: the possible results of a measurement are exactly the eigenvalues of the observable, and right after you get a result, the system is left in the matching eigenstate. If your energy-meter can only ever read certain values, it is because the energy operator has only those eigenvalues. The discrete "menu" of allowed outcomes is nothing but the operator's list of eigenvalues.
Reading the eigenvalue equation out loud
All of that lives in one tidy line, the eigenvalue equation. Do not be put off by symbols — read it as an English sentence and it is almost obvious.
 |ψ⟩ = a |ψ⟩ | | | | | | | +-- ...the SAME state |ψ⟩, | | +----- ...gives back a number a (the eigenvalue) times... | +------------- ...acting on a special state |ψ⟩ (an eigenstate)... +----------------- The operator  (the observable)...
Here |ψ⟩ is just a compact way of writing "the state ψ" (those angle brackets are standard quantum shorthand you will meet properly later). The equation says: the special state ψ is one the operator  leaves pointing the same way, stretched by the factor a. Find every state for which this holds, collect all the matching a's, and you have built the complete list of values that a measurement of  can ever return. That is, quite literally, all there is to predicting *what outcomes are possible*.
But most states are not eigenstates
Here is the part beginners find genuinely startling, so go slowly. A typical state is *not* an eigenstate of the thing you want to measure. It is a blend — a superposition — of several eigenstates at once, like a chord that contains several musical notes rather than a single pure tone. Such a state has no single definite value of the observable *before* you measure. It is not that the value is hidden from you; it genuinely is not settled.
So what happens when you measure such a blend? You still get *one* of the eigenvalues — never an in-between number — but *which* one is a matter of chance. The rule that fixes the odds is the Born rule: the more strongly an eigenstate is represented in the blend, the more likely its eigenvalue is to come up. A note that is loud in the chord is the one you are most likely to hear. And the instant the result appears, the chord collapses to that single note: the system jumps into the matching eigenstate, and a repeat measurement now gives the same answer for sure.
Pull it together. An observable is a measurable quantity, carried by an operator. Its eigenvalues are the only values a measurement can yield, and its eigenstates are the states that own one of those values outright. Real states are usually blends, so a measurement gambles among the eigenvalues with Born-rule odds, then snaps the system onto the eigenstate it landed on. The energy levels you have already met — say, the rungs of a trapped particle — are exactly this story for the energy eigenstates of the energy operator. Next we ask a sharper question: what must be true of an operator for its eigenvalues to be sensible, real measurement readings in the first place?