This is a super simplified, super brief overview of the family dynamics that lead to the War of the Roses, which are covered in the Henry VI plays.
Edward III’s descendants
Edward III had several children, but only four sons are of critical importance. The oldest is Edward, the Black Prince. He died before his father, so his son Richard II inherited the throne from Edward III.
Edward’s third son (the second died young), Lionel of Antwerp, had a son, Edmund Mortimer, and a daughter, Ann Mortimer. She married one of Edward’s grandsons, Richard, who was born to his fifth son, Edmund of Langley, the 1st Duke of York. Ann and Richard have a son, Richard Plantagenet in Henry VI, Part 1.
Edward III’s fourth son was John of Gaunt and he had four sons that we care about. His legitimate son was Henry IV. His illegitimate – and later legitimized – children were the Beaufort’s. John Beaufort is the Duke of Somerset and passes that title on to his children. Henry Beaufort is the Bishop of Winchester and Thomas Beaufort is the Duke of Exeter.
Laying the groundwork…
Richard II was deposed by Henry IV in 1399. This did not sit well with some in the north since he was not the next in line for the throne. Edmund Mortimer, the next logical heir, attempted a coup, but failed and was put on house arrest. So, Henry V ascended to the throne and was generally considered a great king. However, he died when his son Henry VI was just an infant. Henry VI was not a well-liked king and was thought of as weak-willed. He had the backing of his uncles, the Beauforts, and their descendants, who probably wanted to be King themselves.
Richard Plantagenet, the Duke of York, had a dual claim to the throne because of his parents, Ann Mortimer and Richard (see above). Richard then marries Cecily Neville, who was descended from one of John of Gaunt’s daughters. Their son, Edward IV now had an incredibly strong claim to the throne.
He used that claim to challenge Henry VI and ultimately depose him, putting a stop to the fighting until his death. But, that is a fight for another play.