House of Lancaster
The House of Lancaster claimed the throne through John of Gaunt, Duke of Lancaster the second son of Edward III.
When Edward died the throne went to his grandson Richard II the son of the 'Edward the Black Prince'. Richard saw his Lancastrian cousins as a threat and banished Henry Bolingbroke and confiscated his large inheritance. Henry raised an army and forced Richard to abdicate and declared himself Henry IV.
When Edward III had declared his claim to the throne of France he started the Hundred Years War which was still raging through this time. One of the most famous battles was 1415's Battle of Agincourt when Henry V's English army defeated a French army three times their size.
However, when Henry V died he was succeeded by his baby son. Henry VI was a weak king he lost much of the English possessions in France. He became prone to fits of madness and the Wars of the Roses began between the house of York and Lancaster.