Clarence House holds a lot of history. It has housed many of the current British Royal family: from Queen Elizabeth II and her husband Prince Philip to Queen Elizabeth, the Queen Mother (aka the Queen Mum). It's current residents are Prince Charles, Prince of Wales, and his wife Camila, Duchess of Cornwall.
The royal pair took over the house in 2002 (which was built between 1825 and 1827), from Prince Charles' grandmother the Queen Mother.
Since then, the Prince of Wales has redecorated it some, but has left a lot of the pieces the Queen Mother had incorporated into the home during her time there from 1953 to 2002.
Other British Royals that have lived and roamed the Clarence House halls? Prince William lived there from 2003 to 2011, and Prince Harry lived there from 2003 to 2012.
Since 2003, the house that is connected to St. James's Palace has been called "Clarence House," as it is where the office of Prince Charles resides.