We use multifiber spectroscopy of 12 poor groups of galaxies to address (1) whether the groups are bound systems or chance projections of galaxies along the line of sight; (2) why the members of each group have not already merged to form a single galaxy, despite the groups' high galaxy densities, short crossing times, and likely environments for galaxy-galaxy mergers; and (3) how galaxies might evolve in these groups, where the collisional effects of the intragroup gas and the tidal influences of the global potential are weaker than in rich clusters. Each of the 12 groups has fewer than about five cataloged members in the literature. Our sample consists of 1002 galaxy velocities, 280 of which are group members. The groups have mean recessional velocities between 1600 and 7600 km s(-1). Nine groups, including three Hickson compact groups, have the extended X-ray emission characteristic of an intragroup medium (see Paper II). We conclude the following: 1. The nine poor groups with diffuse X-ray emission are bound systems with at least similar to 20-50 group members with absolute magnitudes as faint as M-B similar to -14 + 5 log(10) h to -16 + 5 log(10) h. The large number of group members, the significant early-type population (up to similar to 55% of the membership) and its concentration in the group center, and the correspondence of the central, giant elliptical with the optical and X-ray group centroids argue that the X-ray groups are not radial superpositions of unbound galaxies. The velocity dispersions of the X-ray groups range from 190 to 460 km s(-1). We are unable to determine if the three non-X-ray groups, which have lower velocity dispersions (<130 km s(-1)) and early-type fractions (= 0%), are also bound. 2. Galaxies in each X-ray-detected group have not all merged together because a significant fraction of the group mass lies outside of the galaxies and in a common halo. The velocity dispersion of the combined group sample is constant as a function of radius out to the virial radius of the system (typically similar to 0.5 h(-1) Mpc). The virial mass of each group (similar to 0.5-1 x 10(14) h(-1) M.) is large compared with the mass in the X-ray gas and in the galaxies (e.g., similar to 1 x 10(12) h(-5/2) M. and similar to 1 x 10(13) h(-1) M., respectively, in NGC 533). These results imply that most of the group mass is in a common, extended halo. The small fraction (similar to 10%-20%) of group mass associated with individual galaxies suggests that the rate of galaxy-galaxy interactions is lower than for a galaxy-dominated system, allowing these groups to virialize before all of their galaxies merge and to survive for more than a few crossing times. 3. The position of the giant, brightest elliptical in each X-ray group is indistinguishable from the center of the group potential, as defined by the mean velocity and the projected spatial centroid of the group galaxies. This result suggests that dominant cluster ellipticals, such as cD galaxies, may form via the merging of galaxies in the centers of poor group-like environments. Groups with a central, dominant elliptical may then fall into richer clusters. This scenario explains why cD galaxies do not always lie in the spatial and kinematic center of rich clusters but instead occupy the centers of subclusters in non-virialized clusters. 4. The fraction of early-type galaxies in the poor groups varies significantly, ranging from that characteristic of the field (less than or similar to 25%) to that of rich clusters (similar to 55%). The high early-type fractions are particularly surprising because all of the groups in this sample have substantially lower velocity dispersions (by a factor of similar to 2-5) and galaxy number densities (by a factor of similar to 5-20) than are typical of rich clusters. Hence, the effects of disruptive mechanisms like galaxy harassment on the morphology of poor group galaxies are weaker than in cluster environments. In contrast, the kinematics of poor groups make them preferred sites for galaxy-galaxy mergers, which may alter the morphologies and star formation histories of some group members. If galaxy-galaxy interactions are not responsible for the high early-type fractions, it is possible that the effects of environment are relatively unimportant at the current epoch and that the similarity of the galaxy populations of rich clusters and some poor groups reflects conditions at the time of galaxy formation. 5. The fraction of early-type group members that have experienced star formation within the last similar to 2 h(-1) Gyr is consistent with that in rich clusters with significant substructure (similar to 15%). If some of the subclusters in these rich, complex clusters are groups that have recently fallen into the cluster environment, the similarity between the star formation histories of the early types in the subclusters and of those in our sample of field groups indicates that the cluster environment and associated mechanisms like ram pressure stripping are not required to enhance and/or quench star formation in these particular galaxies. If the recent star formation is tied to the external environment of the galaxies and not to internal instabilities, it is more likely that galaxy-galaxy encounters have altered the star formation histories of some early-type galaxies in groups and in subclusters.
