We have used the ROSAT PSPC to search for diffuse, extended X-ray emission in a sample of 14 galaxy groups. A diffuse component was found in four cases. Combining our new analysis with published X-ray observations, we found that a total of 48 galaxy groups of varying size and velocity dispersion have analyzed ROSAT X-ray data. Diffuse, approximate to 1.0 keV X-ray gas is seen in 25 of the 48 groups analyzed, but the true incidence of such an intragroup medium is probably lower, since this sample includes X-ray-discovered groups. X-ray luminosity does not correlate well with the optical richness, blue luminosity, or velocity dispersion of the group. In contrast, there is a strong correlation with the percentage of early-type (E and S0) galaxies. All of the groups with an extended intragroup medium have high percentages of early-type galaxies, and over half of these systems appear to contain no spirals at all. Furthermore, all the X-ray-detected systems contain at least one elliptical with a blue luminosity of L(B) similar to 5 x 10(10) L(.) or greater. We discuss several possible explanations for the correlation between spiral fraction and the presence of diffuse hot gas, including the idea that the spiral-rich groups represent superpositions of galaxies, that they contain a relatively cool intragroup medium, or that the formation and evolution of spiral-rich groups are fundamentally different from those of elliptical-rich groups. We also use the X-ray observations to estimate the total masses of these systems. There is a large range in the derived gas masses, but in general the mass in the X-ray-emitting gas is comparable to or less than the mass in the galaxies. Despite the large range in gas mass, there is a very narrow range in total group mass, with most of the groups having a mass of similar to 2 x 10(13) M(.) out to the radius for which X-ray emission is detected by ROSAT. The ratio of observed luminous mass (i.e., galaxies + hot gas) to total inferred mass in poor groups is rather low (approximate to 5%-30%), implying that these systems are dominated by dark matter.