Cores are provinces that are considered a rightful part of your realm. Cores are permanent and can only be removed by forming some cultural unions (forming Arabia will remove Nejd's cores), by forcing a nation to release another by casus belli or crisis, or by decision (Treaty of London removing Netherlands' cores on Belgium).
Two or more nations can have cores on the same land, which can make them fight over it. Non-Great Power nations can use national focus points to raise tension in flashpoints (provinces with their cores owned by another nation). Eventually, this will result in a Crisis. If it elicits interest from the GP's, there may be a war or peaceful transfer of provinces. If a country controls another country's cores, nationalist movements will spawn rebels if militancy is too high.
How to gain cores
The generic event that gives cores only fires if you have Nationalism and Imperialism researched and are a Great Power. It will occur only in states that are non-colonial and either have a majority of primary or accepted cultures, or border a core province of the state. The mean time to happen (MTTH) is 10 years. This can be decreased significantly if the nation's primary culture is the majority of the province and if the province has low militancy and consciousness.
Other than the generic event, certain events and decisions give cores like Manifest Destiny