  We study the distribution functions of mass and circular velocity for dark
matter halos in N-body simulations of the $\Lambda$CDM cosmology, addressing
redshift and environmental dependence. The dynamical range enables us to
resolve subhalos and distinguish them from "distinct" halos. The mass function
is compared to analytic models, and is used to derive the more observationally
relevant circular velocity function. The distribution functions in the velocity
range 100--500 km/s are well fit by a power-law with two parameters, slope and
amplitude. We present the parameter dependence on redshift and provide useful
fitting formulae. The amplitudes of the mass functions decrease with z, but,
contrary to naive expectation, the comoving density of halos of a fixed
velocity ~200 km/s actually increases out to z=5. This is because high-z halos
are denser, so a fixed velocity corresponds to a smaller mass. The slope of the
velocity function at z=0 is as steep as ~ -4, and the mass and velocity
functions of distinct halos steepen with increasing z, while the functions of
subhalos do not steepen with $z$, and become even flatter at z>2. A simple
observable prediction is that the slope of the velocity function of isolated
galaxies is steeper than that of galaxies in groups by as much as unity,
reflecting the density biasing of high-velocity halos. We confirm that the
Press-Schechter approximation typically overestimates the halo mass function by
a factor of ~2, while modified approximations provide improved predictions.
