Personal union
A personal union is the combination of two or more states who have the same monarch while their boundaries, laws, and interests remain distinct. It differs from a federation in that each constituent state has an independent government, whereas a federal state is united by a central government. The ruler in a personal union does not need to be a hereditary monarch.
Personal unions can arise for several reasons, ranging from coincidence (a woman who is already married to a king becomes queen regnant, and their child inherits the crown of both countries) to virtual annexation (where a personal union sometimes was seen as a means of preventing uprisings). They can also be codified (i.e., the constitutions of the states clearly express that they shall share the same person as head of state) or non-codified, in which case they can easily be broken (e.g., by the death of the monarch when the two states have different succession laws).
The Commonwealth realms, not addressed in the list of personal unions, are contemporary independent states that share the same person as monarch.