In politics, a loyalist is someone who stays faithful to a party or government even during times of upheaval and revolt. During America's colonial era, loyalists supported British rule.
Today in the U.K., someone who's described as a loyalist is typically a supporter of political union between Northern Ireland and Great Britain. This term is often used during times of upheaval in various countries, to talk about those who remain loyal to an established government. It's been used this way since at least the 17th century — but if your favorite childhood movie gets a terrible remake, feel free to say you're a loyalist to the original version.