the Hamiltonian operator
If you had to name the single most important operator in quantum mechanics, it would be the Hamiltonian. It plays a double role: it is the operator whose values are the system's energies, and it is the generator of time — the object that drives how every state changes moment to moment. Know a system's Hamiltonian and you know, in principle, everything about both its allowed energies and its dynamics.
The Hamiltonian is the operator for total energy, kinetic plus potential, H = T + V = p^2/(2m) + V(x) for one particle. It governs dynamics through the time-dependent Schrodinger equation i hbar d|psi>/dt = H|psi>, whose formal solution is the time-evolution operator |psi(t)> = exp(-i H t / hbar)|psi(0)> when H is time-independent. Its eigenvalue equation H|n> = E_n|n> is the time-independent Schrodinger equation, so its eigenvalues are the allowed energies and its eigenstates are the stationary states. Because H is Hermitian those energies are real and time evolution is unitary (probability is conserved).
Where it comes from and why it matters: the quantum Hamiltonian is obtained from the classical Hamiltonian function H(x, p) by canonical quantization, promoting x and p to operators with [x, p] = i hbar (so p becomes -i hbar d/dx in the position representation). Almost every quantum problem — atoms, molecules, solids, fields — begins by writing down the right Hamiltonian; the art of much of physics is choosing and then approximately diagonalizing it.
For the hydrogen atom H = -(hbar^2 / 2m) laplacian - e^2/(4 pi epsilon_0 r); diagonalizing it yields the energies E_n = -13.6 eV / n^2 and the familiar orbital eigenstates.
Write the Hamiltonian, diagonalize it, and the energy spectrum of the atom appears.
Canonical quantization of a classical H can be ambiguous when x and p are multiplied together (ordering ambiguity), since classically they commute but as operators they do not; a symmetric ordering must be chosen by hand.