Angular momentum plays an exceedingly important role in the theory of atoms, so you need to understand the presented material quite thoroughly.
The main practical importance of the section on two particle states is the Pauli exclusion principle. The thrust of this section is to show how this principle comes from the behavior of wave functions for multiple identical particles.
The hydrogen atom is important in it's own right, but also as the basis of a model for the structure of all atoms. The size of the hydrogen atom and its ground state energy are derived from a combination of classical circular orbit results and the uncertainty principle -- there is a particular circular orbit radius for which the electron kinetic energy derived from the uncertainty principle matches that predicted by circular orbit dynamics. The full state of an electron in the scalar potential of a proton is described by the four quantum numbers n = 1,2,3,... (principal), l = 0,1,2,...,n-1 (magnitude of orbital angular momentum), m = -l,-l+1,...,l (orbital angular momentum orientation), and m_s = -1/2,1/2 (spin orientation).
The periodic table of the elements can be understood by assuming that atoms with multiple electrons take bound states in the hydrogen-like potential resulting from an atomic nucleus with multiple protons. In order for this assumption to work, the electrons are assumed not to interact except via the Pauli exclusion principle. Better results can often be obtained via a slightly modified picture in which inner electrons shield outer electrons from the atomic nucleus.
Atomic spectra come from the assumption that transitions of electrons between different atomic states are associated with the emission and absorption of photons. The energies of these photons are derived from the conservation of energy.