What was the name of the protracted conflict between English kings and France?

The conflict you're thinking of is indeed the Hundred Years War, a series of conflicts waged from 1337 to 1453 between the House of Plantagenet, rulers of the Kingdom of England, and the House of Valois, rulers of the Kingdom of France, for control of the French throne. It wasn't one continuous battle, but rather a series of wars and truces spanning over a century. Key figures like Edward III and Henry V of England, and Joan of Arc on the French side, played significant roles. The war saw the introduction of new military technologies like the longbow, which gave the English an early advantage. While "The English War" or "The France War" are far too generic, and "The Ten Years War" is far too short a period, the Hundred Years War specifically captures this extended period of Anglo-French conflict, shaping the national identities of both countries and leaving a lasting impact on European history.