From
eu2wiki:
A core province of a nation can be defined as a province which, if it is held by another country, the original country has a permanent CB on. This is represented to the player (for the player's country only) by the coat of arms displayed atop that province on the map, when using the political map mode (accessable by clicking on the crossed flags emblem above the mini-map).
When acquired in a war, a core province receives no nationalism revolt risk.
Core provinces also provide more tax income than non-core provinces (except under maximum centralization with tax collectors present, in which circumstance the non-core penalty is cancelled out).
Core provinces also experience one-half of the revolt risk caused by war exhaustion, compared to non-core provinces.
Core provinces are twice as likely to be affected by a successful religious conversion.
Core provinces may be demanded in a peace treaty, at half the usual warscore cost. Even if not occupied, they can be demanded at double the usual warscore cost.