Cold condensations are inferred to occur throughout X-ray-emitting cooling flows. This paper investigates the physical conditions within these clouds. Photoionization by the diffuse continuum produced by the surrounding hot gas heats and ionizes the surface of the cloud to intermediate temperatures (typically 6000K) and low ionization (the main species present are atomic or singly ionized). A thermal front occurs at a depth of roughly 6 x 10(15) cm (which corresponds to a hydrogen column density of similar to 3 x 10(17) Cm-2), where the conditions change over to those similar to the cold phase of the interstellar medium. The gas within is predominantly cold (well below 100 K), molecular, and X-ray-heated. Molecular hydrogen forms via H- in the dust-free conditions expected for gas that has rapidly cooled from X-ray-emitting temperatures. The Lyman-Werner bands of H, become optically thick, and the hydrogen becomes highly molecular. Eventually, the cloud becomes self-shielded as the result of a combination of the photoelectric opacity of atomic carbon and Rayleigh scattering, and carbon monoxide forms. Cooling by rotational transitions of CO brings the temperature of the core of the cloud to that of the cosmic background. We argue that this is the most likely state of any cloud with sufficient column density to be self-shielded from the diffuse X-ray continuum. Fragmentation in this core may produce a population of substellar objects.