[Abridged] Since 1971 observations in X rays of thousands galaxy clusters have uncovered huge amounts of hot baryons filling up the deep gravitational potential wells provided by dark matter (DM) halos with sizes of millions light-years and masses of some 10^15 M_sun. At temperatures T~10^8 K and with average densities of n~1 particle per liter, such baryons add up to some 10^14 M_sun. With the neutralizing electrons, they constitute the best proton-electron plasma in the Universe (Intra Cluster Plasma, ICP). A key physical feature of the ICP is constituted by its good local Thermal equilibrium, and by its overall hydrostatic condition in the DM wells, modulated by entropy. The latter is set up in the cluster center by the initial halo collapse, and is progressively added at the outgrowing cluster boundary by standing shocks in the supersonic flow of intergalactic gas into the DM wells. We review these entropy-based models and discuss their outcomes and predictions concerning the ICP observables in X rays and in microwaves. The results provide a baseline for disentangling a number of additional and intriguing physical processes superposed to the general equilibrium. We cover: the central entropy erosion produced by radiative cooling vs. the intermittent energy inputs mainly due to active galactic nuclei and mergers; outer turbulent support linked with weakening shocks and decreasing inflow through the virial boundary, causing reduced entropy production; the development from high to low entropy levels throughout a typical cluster; perturbations of the equilibrium up to outright disruption due to deep impacts of infalling galaxy groups or collisions with comparable companion clusters; relativistic energy distributions of electrons accelerated during such events, producing extended radio emission by synchrotron radiation, and contributing to non-thermal pressure support for the ICP.

62 pages, 13 figures, 1 table. Accepted by Physics Reports