Rollo and his Viking allies conquered a large region of France and besieged Paris until entering vassalage to Charles the Simple, the king of the West Franks through the Treaty of St.-Claire-sur-Epte. In exchange for homage and fealty, Rollo legally gained the territory he and his Viking allies had previously conquered. The name "Normandy" reflects Rollo's Viking (i.e. Northman, Latin Normanorum) origins.
Rollo and his immediate successors were styled as "Counts of Normandy. Some later medieval sources refer to them by the title dux, a Latin term from which is derived the English word "duke"; however, Rollo's great-grandson Richard II was the first to assuredly be styled "Duke of Normandy". Although certain titles were used interchangeably during this period, the title of "duke" was typically reserved for the highest rank of feudal nobility - those who either who owed homage and fealty directly to kings or who were independent sovereigns primarily distinguished from kings by not having dukes as vassals.
William the Conqueror added the Kingdom of England to his realm in the Norman Conquest of 1066. This created a problematic situation wherein William and his descendants were king in England but a vassal to the king in France. Much of the contention which later arose around the title Duke of Normandy (as well as other French ducal titles during the Angevin period) stems from this fundamentally irreconcilable situation.
After the death of William the Conqueror, his eldest son Robert Curthose became Duke of Normandy while a younger son, William Rufus, became the English king. A generation later, Henry, Duke of Normandy became king of England which again united the titles.
English monarchs made subsequent attempts to reclaim their former continental possessions, particularly during the Hundred Years' War. In addition to claiming to be Duke of Normandy, after Henry V entered the Treaty of Troyes in 1420, English and British monarchs claimed the throne of France itself. During this time, English monarchs included "King of France" near the top of their list of titles and included the Royal Arms of France in their own armorial achievements.
British claims to the whole Duchy of Normandy, the throne of France and other French claims were not abandoned until 1801 when George III and Parliament, in the Act of Union, joined the Kingdom of Great Britain with the Kingdom of Ireland and used the opportunity to drop their French claims. By this time, the monarchy itself had been already been abolished in France since 1792.
Although the British monarchy relinquished claims to continental Normandy and other French claims in 1801, the monarch of the United Kingdom retains the title Duke of Normandy in respect to the Channel Islands. The Channel Islands (except for Chausey under French sovereignty) remain Crown dependencies of the British Crown in the present era. Unlike the Isle of Man, these islands have no specific title pertaining to them. Collectively they use the Loyal Toast in the Channel Islands is La Reine, notre Duc or The Queen, our Duke (or when the monarch is male, The King, our Duke).