My field of study is Shakespeare, rather than English history, but this is what insight I can offer you.

So the general answer to your question revolves around inheritance laws in England. Under the rules of lineage, the eldest son of the monarch would receive the bulk of inheritance (lands, titles, etc). This meant that it was necessary to create titles and secure lands for the younger sons in order to keep them in positions of power. The titles Earl of Lancaster and Duke of York were created for this purpose, but they both originated from the Plantagenet line and happened during different periods of time.

Henry III's younger son, Edmund, was in want of a title, so Henry III created the Earldom of Lancaster for him in 1267. Edmond had two sons: Henry (of Grosmont) and Thomas. Thomas was executed by Edward II in 1322 and Henry inherited the lands and titles in 1345. By this time, Edward III had taken power and "promoted" Henry from Earl of Lancaster to Duke of Lancaster, making him both the 1st Duke of Lancaster and 4th Earl of Lancaster-- at the same time.

Edward III had eight sons, five of whom survived into adulthood. The younger four all needed legitimate titles. To secure their nobility/power, he married them to high-ranking noblewomen. His third son, John (of Gaunt), was married to Blanche, the daughter of Henry of Grosmont. When Henry died in 1361, John inherited the title. This marriage is generally considered to be the start of the second house of Lancaster. Among John of Gaunt's surviving children was Henry, who became Henry IV.

Edward III's youngest son, Edmund, was given the title of Duke of York in 1385 and is considered the "founder" of that house.

Sources/for more information: Pollard, A J. The Wars of the Roses. , 2013. Print. Grummitt, David. A Short History of the Wars of the Roses. London: I.B. Tauris, 2013. Print.