An Urca process is a reaction by which a neutrino is emitted from a baryon. It is thought to be a means by which energy is emitted from white dwarves and neutron stars. This can happen in a number of ways, typically a baryon (e.g., mass number) either absorbing or emitting a lepton (e.g., electron), but also emitting a neutrino during the reaction. The end result is faster cooling.
I've seen Urca process written as URCA Process, but it was apparently named for the Urca Casino in the Urca neighborhood of Rio de Janeiro where early discussions took place, and there seems no reason to treat it like an abbreviation.