Computational quantum chemistry: choosing a level of theory

The goal is to solve $\hat{H}\psi = E\psi$ for the electrons. An exact solution exists only for $\ce{H}$ and $\ce{H2+}$; everywhere else, one approximates. Born-Oppenheimer: the nuclei, far heavier, stay fixed while the electrons adjust. The result is a potential energy surface: its minima are stable structures, its saddle points are transition states.