  The conventional theory of homogeneous and heterogeneous nucleation in a
supersaturated vapor is tested by Monte Carlo simulations of the lattice gas
(Ising) model with nearest-neighbor attractive interactions on the simple cubic
lattice. The theory considers the nucleation process as a slow (quasi-static)
cluster (droplet) growth over a free energy barrier $\Delta F^*$, constructed
in terms of a balance of surface and bulk term of a "critical droplet" of
radius $R^*$, implying that the rates of droplet growth and shrinking
essentially balance each other for droplet radius $R=R^*$. For heterogeneous
nucleation at surfaces, the barrier is reduced by a factor depending on the
contact angle. Using the definition of "physical" clusters based on the
Fortuin-Kasteleyn mapping, the time-dependence of the cluster size distribution
is studied for "quenching experiments" in the kinetic Ising model, and the
cluster size $\ell ^*$ where the cluster growth rate changes sign is estimated.
These studies of nucleation kinetics are compared to studies where the relation
between cluster size and supersaturation is estimated from equilibrium
simulations of phase coexistence between droplet and vapor in the canonical
ensemble. The chemical potential is estimated from a lattice version of the
Widom particle insertion method. For large droplets it is shown that the
"physical clusters" have a volume consistent with the estimates from the lever
rule. "Geometrical clusters" (defined such that each site belonging to the
cluster is occupied and has at least one occupied neighbor site) yield valid
results only for temperatures less than 60% of the critical temperature, where
the cluster shape is non-spherical. We show how the chemical potential can be
used to numerically estimate $\Delta F^*$ also for non-spherical cluster
shapes.
