The Hamiltonian
The Hamiltonian is the single most important object in non-relativistic quantum mechanics. It is an operator. It represents total energy. It generates time evolution. Everything else is commentary.
In classical mechanics, the Hamiltonian $H(q,p)$ is a function on phase space. You write it in terms of generalized coordinates and their conjugate momenta. The Hamiltonian equations give you first-order equations of motion. The Legendre transform relates it to the Lagrangian. This is all classical machinery. Quantum mechanics copies the structure but changes the nature of the objects. The classical variables $q$ and $p$ become operators $\hat{q}$ and $\hat{p}$ satisfying $[\hat{q},\hat{p}] = i\hbar$. The classical Hamiltonian becomes a quantum operator $\hat{H}$. The replacement is called canonical quantization. It is a rule, not a derivation.
For a single particle moving in one dimension under a potential $V(x)$, the quantum Hamiltonian is:
$$\hat{H} = \frac{\hat{p}^2}{2m} + V(\hat{x}) = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)$$
The first term is kinetic energy. The second is potential energy. The kinetic term contains the second derivative because momentum is represented by $-i\hbar\partial/\partial x$. The potential term is multiplicative because position is represented by multiplication. This is the position representation. In the momentum representation, the roles reverse.
The Hamiltonian is Hermitian (self-adjoint). This is not optional. A Hermitian operator has real eigenvalues, which is necessary because eigenvalues correspond to measurable energy values. It also has a complete orthonormal set of eigenvectors, which is necessary because you need to be able to expand any state in the energy basis. The spectrum can be discrete, continuous, or both. Hydrogen has a discrete negative-energy spectrum (bound states) and a continuous positive-energy spectrum (scattering states). The harmonic oscillator has a purely discrete spectrum. The free particle has a purely continuous spectrum. The structure of the spectrum reflects the physics.
The Hamiltonian generates time translations. This is the content of the Schrödinger equation:
$$i\hbar\frac{d}{dt}|\psi\rangle = \hat{H}|\psi\rangle$$
More precisely, $\hat{H}$ is the generator of the one-parameter unitary group $U(t) = e^{-i\hat{H}t/\hbar}$. Stone's theorem guarantees that every strongly continuous one-parameter unitary group has a self-adjoint generator. In quantum mechanics, that generator is the Hamiltonian. This is a mathematical theorem, not an assumption. Time evolution is unitary evolution generated by the Hamiltonian.
Conserved quantities are Hamiltonians for their own symmetries. If the Hamiltonian commutes with some operator $\hat{A}$, then $\hat{A}$ is conserved. If the system is translationally invariant, the Hamiltonian commutes with the momentum operator, and momentum is conserved. If the system is rotationally invariant, the Hamiltonian commutes with angular momentum. If the Hamiltonian does not depend explicitly on time, energy is conserved. These are the quantum versions of Noether's theorem.
The Hamiltonian of a many-body system is written in second quantization by promoting creation and annihilation operators. For fermions, the operators anticommute. For bosons, they commute. The Hamiltonian takes the form:
$$\hat{H} = \sum_{ij} t_{ij}a_i^\dagger a_j + \frac{1}{2}\sum_{ijkl} V_{ijkl}a_i^\dagger a_j^\dagger a_l a_k$$
The first term describes single-particle hopping or kinetic energy. The second describes interactions. This is the starting point for condensed matter physics, quantum chemistry, and quantum field theory. The form is the same. The operators change. The physics changes with them.
Some Hamiltonians can be solved exactly. The free particle, the harmonic oscillator, the hydrogen atom, the rigid rotator. These are the solvable models. They are taught because they are solvable, not because they are realistic. Real systems are messy. They have anharmonicity, spin-orbit coupling, many-body interactions, disorder. Solvable models are reference points. They tell you what to expect when everything simplifies. Deviations from them tell you what the complicating factors do.
The Hamiltonian defines a quantum system. Specify $\hat{H}$ and you have specified the model. Compute its spectrum and you know the possible energies. Compute its eigenstates and you know the stationary configurations. Compute its time evolution and you know the dynamics. The Hamiltonian is the entire theory.