We have investigated the spectral characteristics of a sample of bright gamma-ray bursts detected with the gamma-ray burst sensors aboard the satellite Ginga. This instrument employed a proportional and scintillation counter to provide sensitivity to photons in the 2-400 keV region and as such provided a unique opportunity to characterize the largely unexplored X-ray properties of gamma-ray bursts. The photon spectra of the Ginga bursts are well described by a low-energy slope, a bend energy, and a high-energy slope. In the energy range where they can be compared, this result is consistent with burst spectral analyses obtained from the BATSE experiment aboard the Compton Gamma-Ray Observatory. However, below 20 keV we find evidence for a positive spectral number index in approximately 40% of our burst sample, with some evidence for a strong rolloff at lower energies in a few events. There is a correlation (Pearson's r = -0.62) between the low-energy slope and the bend energy. We find that the distribution of spectral bend energies extends below 10 keV. There has been some concern in cosmological models of gamma-ray bursts (GRBs) that the bend energy covers only a small dynamic range. Our result extends the observed dynamic range, and, since we observe bend energies down to the limit of our instrument, perhaps observations have not yet limited the range. The Ginga trigger range was virtually the same as that of BATSE, yet we find a different range of fit parameters. One possible explanation might be that GRBs have two break energies, one often in the 50-500 keV range and the other near 5 keV. Both BATSE and Ginga fit with only a single break energy, so BATSE tends to find breaks near the center of its energy range, and we tend to find breaks in our energy range. The observed ratio of energy emitted in the X-rays relative to the gamma rays can be much larger than a few percent and, in fact, is sometimes larger than unity. The average for our 22 bursts is 24%. We also investigated spectral evolution in two bursts. In these events we find strong evidence for spectral softening as well as a correlation between photon intensity and spectral hardness. We also find that the X-ray signal below 30 keV itself softens in both of these events. There is one example of a strong X-ray excess at low energy. In addition to providing further constraints on gamma-ray burst models, the description provided here of burst spectra down to 2 keV should prove useful to future planned efforts to detect bursts at X-ray energies.