Why were the Houses of Lancaster and York fighting?

The basic issue was who was the rightful king of England.
Both Lancaster and York families were descendants of the Plantagenet. Both sides claimed
the throne through descent from King Edward III. The nature of their competing claims is
too complicated to trace here; but can be read in the posted link. The so called "Wars
of the Roses" were largely sporadic between 1455 and 1485; but came to a head with the
assassination of the young King Edward V. Richard of Gloucester had acted as the King's
regent during his minority; but had deposed him and assumed the throne for himself as
Richard III. Shortly thereafter, Edward and his younger brother, the Duke of York, were
mysteriously murdered in the Tower of London. The murders remain unsolved to this day
although many historians and writers, including Shakespeare, believe Richard III to be
responsible.

The Lancastrian claim at this time was held by
Henry Tudor a distant Lancaster relative. He defeated Richard at the Battle of Bosworth
Field, and married Elizabeth of York, thus uniting the two houses and ending the
conflict. An excellent discussion of the Wars of the Roses can be found in Allison
Weir's The Wars of the Roses.