Short answers, pulled from the story.
Edmund of Langley founded the House of York in 1385 when he was created the first Duke of York. He was the fourth surviving son of King Edward III and established the cadet branch that would eventually challenge the reigning House of Lancaster.
Richard of York claimed the throne for the House of York in October 1460. This move turned a political struggle into a civil war for the crown and led to his death at the battle of Wakefield on the 30th of December 1460.
The fate of the Princes in the Tower remains a mystery as it is unknown whether they were killed or who might have killed them. Parliament declared in the document Titulus Regius that the two boys were illegitimate on the grounds that Edward IV's marriage was invalid.
The House of York went extinct in the legitimate male line with the death of Edward Plantagenet, 17th Earl of Warwick, on the 28th of November 1499. This execution followed the suppression of the uprising at the Battle of Stoke Field in 1487.
Henry Tudor united the surviving houses of York and Lancaster by taking Elizabeth of York, the eldest child of Edward IV, as his wife. He acceded to the throne as Henry VII, founder of the Tudor dynasty which reigned until 1603.