Perturbation theory is a generalization of a mechanism for orbiting three body problems, by breaking it into two simple two-body orbiting problems along with unsolvable but approximable perturbations (slight changes) of the two orbits, i.e., how each orbit is slightly modified over time by the presence of the other body. As a theory, it has been generalized to handle other unsolvable equations (often differential equations) for other kinds of physical systems, in chemistry, quantum mechanics, etc. The perturbation is cast as a series of ever-smaller solvable equations, e.g., like a Taylor series.
A first order perturbation problem (or regular perturbation problem) is a problem that includes a very small parameter and the solution can be found by approximating that parameter as zero. If such an approximation is too far off, it is known as a second order perturbation problem (or singular perturbation problem or degenerate perturbation problem).
The Hamiltonian is useful in perturbation theory to study secular (long term) motions related to planetary orbits, and is useful in series-expanded form to provide tractable estimates and because it can handle coordinate transformations that simplify solutions, sometimes effectively eliminating a coordinate. With a multipole expansion, terms through the octupole term can be necessary to explain observed exotic extra-solar planet orbits.