We have carried out a detailed abundance analysis using high-dispersion spectra from HIRES at Keck for a sample of 16 carbon stars found among candidate extremely metal-poor (EMP) stars from the Hamburg/ESO Survey (HES). We find that the Fe metallicities for the cooler C stars (Teff ~ 5100 K) have been underestimated by a factor of ~10 by the standard HES tools. The results presented here provided crucial supporting data used recently by Cohen et al. to derive the frequency of C stars among EMP stars. C enhancement in these EMP C stars appears to be independent of Fe metallicity and approximately constant at ~ the solar (C). The C enhancement shows some evidence of decreasing with decreasing Teff (increasing luminosity), presumably due to mixing and dredge-up of C-depleted material. The mostly low 12C/13C ratios (~4) and the high N abundances in many of these stars suggest that material that has been through proton burning via the CN cycle comprises most of the stellar envelope. C enhancement in this sample is associated with strong enrichment of heavy nuclei beyond the Fe peak for 12 of the 16 stars. The remaining C stars from the HES, which tend to be the most Fe-poor, show no evidence for enhancement of the heavy elements. Very high enhancements of lead are detected in some of the C stars with highly enhanced Ba. The strong lead lines, the high Ba/Eu ratios, and the high ratios of abundances of the diagnostic elements in the first and second s-process peaks demonstrate that the s-process is responsible for the enhancement of the heavy elements for the majority of the C stars in our sample. The low 12C/13C ratios and large C and N enhancements of the EMP C stars are more extreme than those of intrinsic asymptotic giant branch C stars of near-solar Fe metallicity, but closer to the composition of CH stars. Our subsample of EMP C stars without s-process enhancement is reminiscent of the R-type C stars in the solar neighborhood; thus, we expect that they are formed by similar mechanisms. We suggest that both the s-process-enhanced and Ba-normal C stars result from phenomena associated with mass transfer in binary systems. This leads directly to the progression from C stars to CH stars and then to Ba stars as the Fe metallicity increases.