We present a detailed investigation into which properties of CDM halos make them effective strong gravitational lenses. Strong-lensing cross sections of 878 clusters from an N-body simulation are measured by ray-tracing through 13,594 unique projections. We measure concentrations, axis ratios, orientations, and the substructure of each cluster, and compare the lensing-weighted distribution of each quantity to that of the cluster population as a whole. The concentrations of lensing clusters are on average 34% larger than the typical cluster in the universe. Despite this bias, the anomalously high concentrations (c > 14) recently measured by several groups appear to be inconsistent with the concentration distribution in our simulations, which predict that <2% of lensing clusters should have concentrations this high. No correlation is found between lensing cross section and substructure. We introduce several types of simplified dark matter halos and use them to isolate which properties of CDM clusters make them effective lenses. The abundance of giant arcs is primarily determined by the mass distribution within an average overdensity of ~10,000. A multiple-lens-plane ray-tracing algorithm is used to show that projections of large-scale structure increase the giant arc abundance modestly, by <7%. We revisit the claimed excess of giant arcs behind high redshift clusters in the RCS survey and find that the number of high-redshift (z > 0.6) lenses is in good agreement with ΛCDM, although our simulations predict more low-redshift (z < 0.6) lenses than observed.