An heir is the person who is next in line for the throne in a monarchy. Since the Heir to the Throne expansion, the game shows the name, stats, claim strength and age of a kingdom's current heir. In earlier editions, a new monarch would be randomly generated the moment the old one died.
If an heir is at least 15 years old when the current monarch dies, the heir will ascend to the throne. If the heir is younger, a regency council will watch over the country until the heir is old enough.
If an heir dies, there is some chance that the second in line (a younger brother, a cousin...) immediately takes their place. Otherwise, the country will go through a period with no legitimate heir, risking a long regency, a dynasty change or perhaps even a personal union under another country.
Republics and theocracies do not have heirs, as their governments are elected. Therefore they can never have a Regency Council.