In both movies the plot revolves around a pair of identical female twins, who are separated in infancy when their parents separate, each taking one twin with them. In the 1961 original, one twin lives in Boston and the other twin in California, while in the 1998 remake one lives in London and the other in California (the change in locations between the movies is arguably a consequence of increasing globalisation). They meet on a summer camp, where they decide to switch places and then plot to reunite their parents (Maureen O'Hara and Brian Keith in the 1961 movie).
Both twins are played by one actor (Hayley Mills in the 1961 version, Lindsay Lohan in the 1998 version), although this is managed much better in the 1998 version, through the use of computer technology.