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An Urca process is a particular reaction in which a neutrino is emitted from a baryon. It is thought to be a means by which energy is emitted from white dwarfs and neutron stars. The process can happen in a number of ways, typically a baryon (e.g., nucleon) either absorbing or emitting a lepton (e.g., electron), emitting a neutrino during the reaction. The end result is faster cooling (Urca-process cooling) than otherwise, given the number of neutrinos that immediately escape the body.
Under certain conditions (high temperature and density and certain ratios of constituents), the direct Urca process (dUrca) can take place, but under less extreme conditions, the modified Urca process (mUrca) can still take place, in which two nucleons rather than one start the process. The latter produces much less cooling: a smaller fraction of the energy ends up with the neutrino. For this reason, for a short time at the beginning of its life, a neutron star is expected to cool very quickly until the conditions for dUrca are no longer met.
Within white dwarfs (and the internal shells of massive post-main-sequence stars?) an Urca shell might form, a shell-shaped region where the Urca process leads to cooling. Thermal Urca is the Urca process in a "reaction loop" (possibly including other reactions) that overall, saps the matter's heat to fuel the process. Convective Urca refers to a situation where the cooling of the Urca process contributes to convection, the Urca process occurring at certain radii.
Regarding capitalization of Urca, the original theorists, Gamow and Schoenberg, first spelled it urca process in their 1941 paper. It is now also sometimes written as the Urca process and sometimes URCA process. The name was apparently inspired by the Urca Casino in the Urca neighborhood of Rio de Janeiro where early discussions took place. The paper was submitted with no explanation of the letter-string (which fact is attributed to Gamow's sense of humor) and Gamow had a backronym phrase chosen in case the editor asked: "un-recordable cooling agent".