We study dark-matter halo density profiles in a high-resolution N-body simulation of an LCDM cosmology. Our statistical sample contains ~5000 haloes in the range 10^{11}-10^{14} M_sun. The profiles are parameterized by an NFW form with two parameters, an inner radius r_s and a virial radius r_v, and we define the halo concentration c_v = r_v/r_s. First, we find that, for a given halo mass, the redshift dependence of the median concentration is c_v ~ 1/(1+z), corresponding to a roughly constant r_s with redshift. We present an improved analytic treatment of halo formation that fits the measured relations between halo parameters and their redshift dependence. The implications are that high-redshift galaxies are predicted to be more extended and dimmer than expected before. Second, we find that the scatter in log(c_v) is ~0.18, corresponding to a scatter in maximum rotation velocities of dV/V ~ 0.12. We discuss implications for modelling the Tully-Fisher relation, which has a smaller reported intrinsic scatter. Third, haloes in dense environments tend to be more concentrated than isolated haloes. These results suggest that c_v is an essential parameter for the theory of galaxy modelling, and we briefly discuss implications for the universality of the Tully-Fisher relation, the formation of low surface brightness galaxies, and the origin of the Hubble sequence.
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