an observable
An observable is any physical quantity you can actually measure — position, momentum, energy, a component of angular momentum, spin. The move that defines quantum mechanics is that each such quantity is not a number attached to the system but an operator acting on its state vector. The measurable numbers are hidden inside that operator, waiting to be extracted by measurement.
Formally, an observable is represented by a Hermitian operator A. Its eigenvalues a_n are the only possible results of a measurement, and its eigenstates |a_n> are the states in which that result is certain. Measuring A on a state |psi> yields the outcome a_n with probability |<a_n|psi>|^2 (the Born rule) and leaves the system in the corresponding eigenstate |a_n> afterward. Because A is Hermitian those outcomes are real and the eigenstates form a complete basis, so the possible-values-and-probabilities story is always well defined.
The organizing idea is compatibility. Two observables can be measured simultaneously to arbitrary precision if and only if their operators commute, [A, B] = 0, in which case they share a common eigenbasis; if they do not commute, they obey an uncertainty relation and cannot both be sharp. A maximal set of mutually commuting observables — a complete set of commuting observables, or CSCO — provides a full set of labels (quantum numbers) that uniquely name every basis state, which is how, for instance, an atomic state is tagged by n, l, m, and spin.
Energy is the observable represented by the Hamiltonian H; position x and momentum p are observables that do not commute ([x, p] = i hbar), so no state has both perfectly defined.
Commuting observables share sharp values; non-commuting ones trade sharpness via the uncertainty relation.
Not every Hermitian operator corresponds to something one routinely measures, and famously time is a parameter, not an observable operator, in ordinary quantum mechanics.