Perhaps it would be best to give a relatively brief overview of the House of Plantagenet. It came to rule the Kingdom of England towards the end of 1154 due to the following reasons:
1) King Henry I of England (r. 1100 - 1135) had had a lot of illegitimate children, but from his legitimate ones, only Matilda and William survived into adulthood. As the latter predeceased his father (without leaving any issue), the King made his barons accept Matilda as his eventual successor. This they did, but once he was out of the picture, they proclaimed his nephew, Stephen, to be their new ruler.
2) In 1128, Matilda married Geoffrey, Count of Anjou, a member of what we now call the House of Plantagenet.
3) Before his death in 1154, King Stephen had decided to be succeeded by Matilda's son, Henry.
So we have:
1. King Henry II (r. 1154 - 1189). His wife was Eleanor of Aquitaine, who gave birth to his successor...
2. King Richard I (r. 1189 - 1199). His wife, Berengaria of Navarre, did not bear him any issue. Due to the fact that the laws of succession had not been set in stone yet, he was succeeded not by his nephew Arthur I, Duke of Brittany, but by his brother...
3. King John (r. 1199 - 1216). His wife, Isabella of Angouleme, gave birth to his successor...
4. King Henry III (r. 1216 - 1272). His wife, Eleanor of Provence, gave birth to his successor...
5. King Edward I (r. 1272 - 1307). His wife, Eleanor of Castile, gave birth to his successor...
6. King Edward II (r. 1307 - 1327). His wife, Isabella of France, gave birth to his successor...
7. King Edward III (r. 1327 - 1377). His eldest son, Edward, the Black Prince, had predeceased him, so he was succeeded by his grandson...
8. King Richard II (r. 1377 - 1399). He did not have any issue and was made to abdicate in favour of his cousin from the House of Lancaster (agnatic primogeniture)...
9. King Henry IV (r. 1399 - 1413). His wife, Mary de Bohun, gave birth to his successor...
10. King Henry V (r. 1413 - 1422). His wife, Catherine of Valois, gave birth to his successor...
11. King Henry VI (r. 1422 - 1461 and 1470 - 1471). He was deposed (twice) by his third cousin (agnatically) from the House of York...
12. King Edward IV (r. 1461 - 1470 and 1471 - 1483). His wife, Elizabeth Woodville, gave birth to his successor...
13. King Edward V (r. 1483). He and his siblings were all declared illegitimate by his paternal uncle who replaced him as...
14. King Richard III (r. 1483 - 1485). He was defeated by Henry Tudor at the Battle of Bosworth Field, thus putting an end to the rule of the House of Plantagenet.
Edward Plantagenet, 17th Earl of Warwick, was the last legitimate male of the House of Plantagenet (York). He was executed on the orders of King Henry VII in 1499. His elder sister, Margaret Pole, Countess of Salisbury, shared his fate in 1541, thanks to her first cousin-once-removed, King Henry VIII.