The bright GRB 050408 was localized by HETE-2 near local midnight in the Western Hemisphere, enabling an impressive ground-based follow- up effort, as well as space-based follow-up from Swift. The Swift data from the XRT and our own optical photometry and spectrum of the afterglow provide the cornerstone for our analysis. Under the traditional assumption that the visible wave band was above the peak synchrotron frequency and below the cooling frequency, the optical photometry of 0.03-5.03 days shows an afterglow decay corresponding to an electron energy index of p$_lc$=2.05+/-0.04, without a jet break as suggested by others. A break is seen in the X-ray data at early times (at åisebox-0.5ex 12,600 s after the GRB). The spectral slope of the optical spectrum is consistent with p$_lc$ assuming a host galaxy extinction of A$_V$=1.18 mag. The optical-NIR broadband spectrum is also consistent with p=2.05 but prefers A$_V$=0.57 mag. The X-ray afterglow shows a break at 1.26×10$^4$ s, which may be the result of a refreshed shock. This burst stands out in that the optical and X-ray data suggest a large H I column density of N$_HI$i̊sebox-0.5ex 10$^22$ cm$^-2$; it is very likely a damped Lyα system, so the faintness of the host galaxy (M$_V$>-18 mag) is noteworthy. Moreover, we detect extraordinarily strong Ti II absorption lines with a column density through the GRB host that exceeds the largest values observed for the Milky Way by 1 order of magnitude. Furthermore, the Ti II equivalent width is in the top 1% of Mg II absorption-selected QSOs. This suggests that the large-scale environment of GRB 050408 has significantly lower Ti depletion than our Galaxy and a large velocity width (δv>150 km s$^-1$).