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High-resolution N-body simulations are used to examine the power spectrum dependence of the concentration of galaxy-sized dark matter halos. It is found that dark halo concentrations depend on the amplitude of mass fluctuations as well as on the ratio of power between small and virial mass scales. This finding is consistent with the original results of Navarro, Frenk, and White (NFW) and allows their model to be extended to include power spectra substantially different from cold dark matter (CDM). In particular, the single-parameter model presented here fits the concentration dependence on halo mass for truncated power spectra, such as those expected in the warm dark matter scenario, and predicts a stronger redshift dependence for the concentration of CDM halos than proposed by NFW. The latter conclusion confirms recent suggestions by Bullock and coworkers, although this new modeling differs from theirs in detail. These findings imply that observational limits on the concentration, such as those provided by estimates of the dark matter content within individual galaxies, may be used to constrain the amplitude of mass fluctuations on galactic and subgalactic scales. The constraints on LambdaCDM models posed by the dark mass within the solar circle in the Milky Way and by the zero point of the Tully-Fisher relation are revisited, with the result that neither data set is clearly incompatible with the ``concordance'' (Omega0=0.3, Lambda0=0.7, sigma8=0.9) LambdaCDM cosmogony. This conclusion differs from that reached recently by Navarro and Steinmetz, a disagreement that can be traced to inconsistencies in the normalization of the LambdaCDM power spectrum used in that work.