operator
An operator is a mathematical action that you apply to a quantum state and get another state back. In quantum mechanics, every physical quantity you might measure — position, momentum, energy — is represented not by a number but by an operator, usually written with a hat, such as the position operator or the energy operator. You can think of it as a rule that takes a state in and hands a transformed state out.
Most operators of interest are linear, meaning they respect addition and scaling: acting on a sum of two states gives the sum of the results, and doubling the input doubles the output. This linearity is what lets superpositions survive intact when an operator acts, and it is the reason quantum theory is built on the mathematics of vectors and matrices rather than ordinary arithmetic.
Operators do not commute the way numbers do: applying one and then another can give a different result from doing them in the opposite order. This is not a quirk of notation but a genuine statement about the world, and it is the root of uncertainty and of much of what makes quantum mechanics strange. An operator only becomes a measurable quantity once it has the special property of being Hermitian.
An operator acts on a state and returns a new state; physical quantities are represented this way.
An operator is not itself a number you read off a meter. It is the rule that, together with the state, predicts the possible measurement results and their probabilities. Only Hermitian operators correspond to genuine observables.