Type Ia Supernovae (SNe Ia) are excellent distance indicators, yet the full details of the underlying physical mechanism giving rise to these dramatic stellar deaths remain unclear. As large samples of cosmological SNe Ia continue to be collected, the scatter in brightnesses of these events is equally affected by systematic errors as statistical. Thus we need to understand the physics of SNe Ia better, and in particular we must know more about the progenitors of these SNe so that we can derive better estimates for their true intrinsic brightnesses. The host galaxies of SNe Ia provide important indirect clues as to the nature of SN Ia progenitors. In this Thesis we utilize the host galaxies of SNe Ia discovered by the Nearby Supernova Factory (SNfactory) to pursue several key investigations into the nature of SN Ia progenitors and their effects on SN Ia brightnesses. We first examine the host galaxy of SN 2007if, an important member of the subclass of SNe Ia whose extreme brightnesses indicate a progenitor that exceeded the canonical Chandrasekhar-mass value presumed for normal SNe Ia, and show that the host galaxy of this SN is composed of very young stars and has extremely low metallicity, providing important constraints on progenitor scenarios for this SN. We then utilize the full sample of SNfactory host galaxy masses (measured from photometry) and metallicities (derived from optical spectroscopy) to examine several global properties of SN Ia progenitors: (i) we show that SN Ia hosts show tight agreement with the normal galaxy mass-metallicity relation; (ii) comparing the observed distribution of SN Ia host galaxy masses to a theoretical model that couples galaxy physics to the SN Ia delay time distribution (DTD), we show the power of the SN Ia host mass distribution in constraining the SN Ia DTD; and (iii) we show that the lack of ultra-low metallicities in the SNfactory SN Ia host sample gives provisional support for the theorized low-metallicity inhibition of SNe Ia. Finally we revisit recent studies which found that the corrected brightnesses of SNe Ia (after application of the standard light curve width and color corrections) correlate with the masses of their host galaxies. We confirm this trend with host mass using SNfactory data, and for the first time confirm that an analogous trend exists with host metallicity. We then apply a spectroscopic standardization technique developed by SNfactory and show that this method significantly reduces the observed bias. In this Thesis we show that SN Ia host galaxies continue to provide key insight into SN Ia progenitors, and also illuminate possible biases in SN Ia brightness standardization techniques.

We show empirically that fits to the color-magnitude relation of Type Ia supernovae after optical maximum can provide accurate relative extragalactic distances. We report the discovery of an empirical color relation for Type Ia light curves: During much of the first month past maximum, the magnitudes of Type Ia supernovae defined at a given value of color index have a very small magnitude dispersion; moreover, during this period the relation between B magnitude and B-V color (or B-R or B-I color) is strikingly linear, to the accuracy of existing well-measured data. These linear relations can provide robust distance estimates, in particular, by using the magnitudes when the supernova reaches a given color. After correction for light curve stretch factor or decline rate, the dispersion of the magnitudes taken at the intercept of the linear color-magnitude relation are found to be around 0^m .08 for the sub-sample of supernovae with (B_max - V_max) ?= 0^m 0.5, and around 0^m.11 for the sub-sample with (B_max - V_max) ?= 0^m .2. This small dispersion is consistent with being mostly due to observational errors. The method presented here and the conventional light curve fitting methods can be combined to further improve statistical dispersions of distance estimates. It can be combined with the magnitude at maximum to deduce dust extinction. Theslopes of the color-magnitude relation may also be used to identify intrinsically different SN Ia systems. The method provides a tool that is fundamental to using SN Ia to estimate cosmological parameters such as the Hubble constant and the mass and dark energy content of the universe.