Why is Westminster Abbey called Westminster Abbey?

Abbey and minster are synonymous titles that indicate that a church was used by monks. The Abbey is in Westminster and so is called Westminster Abbey. Westminster itself is a borough of Greater London, but was once a separate town. It was called Westminster because it was west of London, and had a minster (the Abbey). London has since grown so big that Westminster is part of London, but the historic area where London was originally founded is still called 'The City of London' or just 'The City' for short.
Westminster Abbey is often thought to be a cathedral due to its size and grandeur. However the cathedral for the Diocese of London is St. Paul's Cathedral.
Neither of these should be confused with Westminster Cathedral, a nearby Roman Catholic cathedral.