The turn of the century is the transition from one century to another, or the time period before or after that change in centuries.
According to the Chicago Manual of Style, there is no agreement as to the meaning of the phrase "turn of the n-th century." For instance, if a statement describes an event as taking place "at the turn of the 18th century," it could refer to a period around the year 1701 or around 1800, that is, the beginning or end of that century. As a result, they recommend either using only "turn of the century," and only in a context that makes clear which transition is meant, or alternatively to use a different expression that is unambiguous.
"Turn of the century" once meant the transition from the 19th century to the 20th century; however, as the generations living at the end of the 20th century survived into the 21st century, it became necessary to indicate the century to avoid confusion.