This dissertation focuses on understanding two different, but in each case extreme, astrophysical environments: the Crab Nebula and emission line galaxies. These rela- tively local objects are well constrained by observations and are test cases of phenom- ena seen at high-z where detailed observations are rare. The tool used to study these objects is the plasma simulation code known as Cloudy. The introduction provides a brief summary of relevant physical concepts in nebular astrophysics and presents the basic features and assumptions of Cloudy. The first object investigated with Cloudy, the Crab Nebula, is a nearby supernova remnant that previously has been subject to photoionization modeling to reproduce the ionized emission seen in the nebula's filamentary structure. However, there are still several unanswered questions: (1) What excites the H2 emitting gas? (2) How much mass is in the molecular com- ponent? (3) How did the H2 form? (4) What is nature of the dust grains? A large suite of observations including long slit optical and NIR spectra over ionized, neutral and molecular gas in addition to HST and NIR ground based images constrain a particularly bright region of H2 emission, Knot 51, which exhibits a high excitation temperature of ∼3000 K. Simulations of K51 revealed that only a trace amount of H2 is needed to reproduce the observed emission and that H2 forms through an uncommon nebular process known as associative detachment. The final chapters of this dissertation focus on interpreting the narrow line region (NLR) in low-z emission line galaxies selected by a novel technique known as mean field independent component analysis (MFICA). A mixture of starlight and radiation from an AGN excites the gas present in galaxies. MFICA separates galaxies over a wide range of ionization into subsets of pure AGN and pure star forming galaxies allowing simulations to reveal the properties responsible for their observed variation in ionization. Emission line ratios can constrain the spectral energy distribution, excitation mechanism, abundances and physical conditions present in these galaxies, while the large data set allows many weaker emission lines to be used as consistency checks. By integrating over a wide range of densities and radii from the excitation source, the variation in ionization for AGN can be represented as change in the central concentration of clouds in the NLR. Preliminary analysis from modeling star forming galaxies indicates that the same interpretation might apply to galaxies without an AGN in which gas is excited by starlight.
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