During the Spitzer Space Telescope cryogenic mission, Guaranteed Time Observers, Legacy Teams, and General Observers obtained Infrared Spectrograph (IRS) observations of hundreds of debris disk candidates. We calibrated the spectra of 571 candidates, including 64 new IRAS and Multiband Imaging Photometer for Spitzer (MIPS) debris disks candidates, modeled their stellar photospheres, and produced a catalog of excess spectra for unresolved debris disks. For 499 targets with IRS excess but without strong spectral features (and a subset of 420 targets with additional MIPS 70{mu}m observations), we modeled the IRS (and MIPS data) assuming that the dust thermal emission was well-described using either a one- or two-temperature blackbody model. We calculated the probability for each model and computed the average probability to select among models. We found that the spectral energy distributions for the majority of objects (~66%) were better described using a two-temperature model with warm (T_gr_~100-500K) and cold (T_gr_~50-150K) dust populations analogous to zodiacal and Kuiper Belt dust, suggesting that planetary systems are common in debris disks and zodiacal dust is common around host stars with ages up to ~1Gyr. We found that younger stars generally have disks with larger fractional infrared luminosities and higher grain temperatures and that higher-mass stars have disks with higher grain temperatures. We show that the increasing distance of dust around debris disks is inconsistent with self-stirred disk models, expected if these systems possess planets at 30-150AU. Finally, we illustrate how observations of debris disks may be used to constrain the radial dependence of material in the minimum mass solar nebula.