Who were the Tudors?

They are Britain's most famous royal dynasty, and yet the House of Tudor was one which should never have sat on the throne of England in the first place. Their lineage was noble, but by no means as grand as many other families in England, and yet they managed to eventually succeed in displacing the house of Plantagenet, who had ruled England for over 300 years. So what actually was their lineage? How did they descend from King Edward III, did this give them the right to wear the crown? How did the name of "Tudor" actually make it's way into English history?