Abdication is the formal act of stepping down from something, especially a king giving up the throne. An abdication is a type of resignation.
When a king — or another person in power — gives up that position, they abdicate. Such an act is then called an abdication. The roots of this word mean declare in Latin, and that's an important part of the meaning. Abdicating isn't just quitting: it's formally quitting, including a declaration of quitting. There’s no such thing as a private abdication. After an abdication, there's a vacuum of power, and there may be a power struggle to fill the empty position.