A dynasty is a family or extended family which retains political power across generations, or more generally, any organization which extends dominance in its field even as its particular members change.
China was ruled by a series of dynasties dating back to the Xia until the Revolution of 1912.
Though in democratic governments rule does not pass automatically by inheritance, political power often accrues to generations of related individuals; see, for example, the list of U.S. political families, which includes the Kennedys, Bushes, and Adamses.
In its more general sense, the term is often applied to sports teams, though sometimes the period of dominance is measured in years, not decades or centuries as with politics. John Wooden led a college basketball dynasty at UCLA in the 1960s and 1970s. In modern professional sports, it usually only takes a three-peat to qualify for dynasty status in the typical hyperbole of sports journalism. The Los Angeles Lakers are a recent example.
A better example in sports might be the New York Renaissance, one of the few cases when an entire team was inducted into the Basketball Hall of Fame.