The term dense core (in discussions of star formation) to refer to a dense molecular cloud or a dense region of a molecular cloud, which would be a necessary stage in the development of stars from clouds. Current star formation models theorize them, and apparent examples are observed, e.g., in well-known clouds in Orion. The term refers to a mass of molecular hydrogen that is gravitationally bound or very close to that state, i.e., the Jeans criterion is met.
A coherent dense core is such a region that is relatively calm, i.e., has thermal kinetic energy but lacks the degree of turbulence evident in the region around it. Current research aims to understand their formation and evolution into stars.
As far as I can tell, in the context of star formation, the word core (without the word "dense") can refer to a dense core, or (by context) can refer to the core of a subsequent protostar.