New intermediate-resolution spectroscopy for certain members of a sample of 68 moderate- to high-redshift QSOs are presented. These new data have been obtained to supplement previously published lower resolution survey data, which we refer to as the Lick survey, in order to identify H I disk galaxies with large redshifts that give rise to absorption lines in the spectra of background QSOs. Data have been obtained with the MMT and the CTIO 4 m telescope for six QSOs at 1.0-2.4 A resolution. Absorption-line lists are presented, and evidence supporting the identification of seven damped Lyα absorption systems in the six QSOs is discussed. Six of the seven damped Lyα systems have neutral hydrogen column densities N(H I) >= 2 x 10^20^ cm^-2^. The damped Lyα systems have absorption properties which include those expected when high-redshift galaxy disks cosmologically intervene along our sight line to QSOs. Therefore, these systems are sometimes termed "Lyα disk" systems. The properties of the damped Lyα systems are discussed and contrasted with other QSO absorption-line systems. About half of the damped Lyα systems would have been missed if they had been searched for on the basis of a spectroscopic survey for C IV absorption alone. Damped Lyα systems are distinguished because they have low ions (i.e., low ionization species) which are generally stronger than the high ions (i.e., high ionization species), they sometimes have other metal line systems clustering around them, and they are the only systems which exhibit 21 cm absorption. In order to investigate these systems further, a standard curve-of-growth analysis was performed when possible. We suggest that damped Lyα systems arise when a sight line intercepts a high-redshift protogalaxy disk containing a quiescent cloud component characterized by high column density and low effective velocity dispersion. At the same time, the sight line usually intercepts a broader turbulent component, which we identify as the halo, characterized by much lower column density and higher effective velocity dispersion. The turbulent component may contain both low and high ions but about half the time only the low ions are strong enough to be detected. In two of the damped Lyα systems only a quiescent component, with effective velocity dispersion σ < 10 km s^-1^, appears to be present. Based on the restrictive assumption that these are single quiescent components, abundances between 4 x 10^-3^ and 1 times solar values are derived. These determinations should be considered strictly lower limits, provided that our metal line identifications and measurements (sometimes in the presence of Lyα forest absorption) are substantially correct. A summary of the incidence of damped Lyα absorption in the Lick survey sample of 68 QSOs is also presented. At least 16 damped Lyα systems with column densities N(H I) >= 2 x 10^20^ are found over a cosmological path length {DELTA}z = 56 (dN/dz >= 0.29 +/- 0.07) at a mean redshift z = 2.24.