This thesis presents a new multi-component spectral energy distribution (SED) decomposition method for analyzing the ultraviolet to millimeter wavelength SEDs of dusty infrared-luminous galaxies. SEDs are constructed from spectroscopic and photometric data obtained with the Spitzer Space Telescope in conjunction with photometry from the literature. Each SED is decomposed into emission from populations of stars, an AGN accretion disk, PAH molecules, atomic and molecular lines, and distributions of graphite and silicate grains. Decompositions of the SEDs of ~20 template starburst galaxies and ~10 template AGNs provide baseline properties to aid in quantifying the star-formation and accretion contributions in composite sources. As case studies, we show that obscured radiation from stars is capable of powering the total dust emission from the composite galaxy NGC6240, while the decomposition of Mrk1014 is consistent with ~65% of its power emerging from an AGN and ~35% from star-formation. The properties of our template starburst galaxies and AGNs are used to estimate the AGN-strengths for some of our ~75 ULIRGs, although many are so obscured that uncovering their principal energy source is not possible in the infrared, and instead will require dust penetrating X-ray observations. We suggest that many of the variations in the SEDs of starburst galaxies may be explained in terms of the different mean optical depths through the clouds of dust surrounding the young stars within each. We also suggest that the divergent far-IR properties of AGNs may result from differences in the relative orientation of their host galaxy and AGN accretion disks, resulting in different amounts of AGN-heated cold dust emission emerging from their host galaxies. We estimate that 30-50% of the far-IR and PAH emission from Mrk1014 may originate from such AGN-heated material in its host galaxy disk, demonstrating that caution must be used when estimating the strength of star-formation based solely upon the far-IR or PAH properties of a source.