Owing to their utility for measurements of cosmic acceleration, Type Ia supernovae (SNe Ia) are perhaps the best-studied class of SNe, yet the progenitor systems of these explosions largely remain a mystery. A rare subclass of SNe Ia shows evidence of strong interaction with their circumstellar medium (CSM), and in particular, a hydrogen-rich CSM; we refer to them as SNe Ia-CSM. In the first systematic search for such systems, we have identified 16 SNe Ia-CSM, and here we present new spectra of 13 of them. Six SNe Ia-CSM have been well studied previously, three were previously known but are analyzed in depth for the first time here, and seven are new discoveries from the Palomar Transient Factory. The spectra of all SNe Ia-CSM are dominated by Hα emission (with widths of åisebox-0.5ex 2000 km s$^-1$) and exhibit large Hα/Hβ intensity ratios (perhaps due to collisional excitation of hydrogen via the SN ejecta overtaking slower-moving CSM shells); moreover, they have an almost complete lack of He I emission. They also show possible evidence of dust formation through a decrease in the red wing of Hα 75-100 days past maximum brightness, and nearly all SNe Ia-CSM exhibit strong Na I D absorption from the host galaxy. The absolute magnitudes (uncorrected for host-galaxy extinction) of SNe Ia-CSM are found to be -21.3 mag <= M$_R$ <= -19 mag, and they also seem to show ultraviolet emission at early times and strong infrared emission at late times (but no detected radio or X-ray emission). Finally, the host galaxies of SNe Ia-CSM are all late-type spirals similar to the Milky Way, or dwarf irregulars like the Large Magellanic Cloud, which implies that these objects come from a relatively young stellar population. This work represents the most detailed analysis of the SN Ia-CSM class to date.