The stellar core is the middle of the star. For a main sequence star the term is generally used to indicate the region where fusion is taking place, or where most of the fusion is taking place. The boundary of what is cited as the "core" appears to be at the radius where there is a significant transition from the conditions in the center of the star. These center-conditions depend on the size of the star.
For small stars such as red dwarves, the core is the location of the fusion, specifically proton-proton chain reactions, which occur at the temperature produced by the small star's gravity.
Larger stars achieve a hotter temperature which triggers the CNO cycle, thus the "transition" from the central conditions is to a surrounding area that lacks the CNO cycle, but which is still sufficiently hot to trigger some proton-proton chain fusion. So the star's core can be described as including the bulk of the star's fusion, but not all.
For post-main sequence giant stars, the conditions of the center are different, and for protostars, pretty much the definition is that there is not yet fusion. Thus the "core" of these would not be distinguished by the characteristics used for main sequence stars.
The solar core is the stellar core of the Sun.