  We present a detailed description of the Voronoi Tessellation (VT) cluster
finder algorithm in 2+1 dimensions, which improves on past implementations of
this technique. The need for cluster finder algorithms able to produce reliable
cluster catalogs up to redshift 1 or beyond and down to $10^{13.5}$ solar
masses is paramount especially in light of upcoming surveys aiming at
cosmological constraints from galaxy cluster number counts. We build the VT in
photometric redshift shells and use the two-point correlation function of the
galaxies in the field to both determine the density threshold for detection of
cluster candidates and to establish their significance. This allows us to
detect clusters in a self consistent way without any assumptions about their
astrophysical properties. We apply the VT to mock catalogs which extend to
redshift 1.4 reproducing the $\Lambda$CDM cosmology and the clustering
properties observed in the SDSS data. An objective estimate of the cluster
selection function in terms of the completeness and purity as a function of
mass and redshift is as important as having a reliable cluster finder. We
measure these quantities by matching the VT cluster catalog with the mock truth
table. We show that the VT can produce a cluster catalog with completeness and
purity $>80%$ for the redshift range up to $\sim 1$ and mass range down to
$\sim 10^{13.5}$ solar masses.
