The Cluster Lensing And Supernova survey with Hubble (CLASH) is a 524-orbit Multi-Cycle Treasury Program to use the gravitational lensing properties of 25 galaxy clusters to accurately constrain their mass distributions. The survey, described in detail in this paper, will definitively establish the degree of concentration of dark matter in the cluster cores, a key prediction of structure formation models. The CLASH cluster sample is larger and less biased than current samples of space-based imaging studies of clusters to similar depth, as we have minimized lensing-based selection that favors systems with overly dense cores. Specifically, 20 CLASH clusters are solely X-ray selected. The X-ray-selected clusters are massive (kT>5keV) and, in most cases, dynamically relaxed. Five additional clusters are included for their lensing strength ({theta}_Ein_>35" at z_s_=2) to optimize the likelihood of finding highly magnified high-z (z>7) galaxies. A total of 16 broadband filters, spanning the near-UV to near-IR, are employed for each 20-orbit campaign on each cluster. These data are used to measure precise ({sigma}_z_~0.02(1+z)) photometric redshifts for newly discovered arcs. Observations of each cluster are spread over eight epochs to enable a search for Type Ia supernovae at z>1 to improve constraints on the time dependence of the dark energy equation of state and the evolution of supernovae. We present newly re-derived X-ray luminosities, temperatures, and Fe abundances for the CLASH clusters as well as a representative source list for MACS1149.6+2223 (z=0.544).