  The number of clusters per site $n(p)$ in percolation at the critical point
$p = p_c$ is not itself a universal quantity---it depends upon the lattice and
percolation type (site or bond). However, many of its properties, including
finite-size corrections, scaling behavior with $p$, and amplitude ratios, show
various degrees of universal behavior. Some of these are universal in the sense
that the behavior depends upon the shape of the system, but not lattice type.
Here, we elucidate the various levels of universality for elements of $n(p)$
both theoretically and by carrying out extensive studies on several two- and
three-dimensional systems, by high-order series analysis, Monte-Carlo
simulation, and exact enumeration. We find many new results, including precise
values for $n(p_c)$ for several systems, a clear demonstration of the
singularity in $n''(p)$, and metric scale factors. We make use of the matching
polynomial of Sykes and Essam to find exact relations between properties for
lattices and matching lattices. We propose a criterion for an absolute metric
factor $b$ based upon the singular behavior of the scaling function, rather
than a relative definition of the metric that has previously been used.
