The arms of England and France are two of the oldest and most well-known in heraldry, so we'll start off by taking a look at their origins.
The Leopards of England
The arms of King Richard I after 1195. These are still the arms of England.
King Richard I "the Lionheart", the famous crusader was the first of the English kings to bear the three golden lions on a red background which remain the arms of England to this day. In early heraldry, the lion was always shown in the position we would now call rampant. When it was found in the position which is now called passant guardant, it was thought to be a completely different animal- a leopard (which was believed to be the offspring of a Lion and a Pard). Thus when King Richard I adopted the three lions (passant) as his arms in 1195, they were called leopards rather than lions.
The arms attributed to King Henry II of England (and to William the Conqueror)
There has been a lot of speculation about what arms were borne by the English kings before Richard I. It was Richard's grandfather, Geoffrey Plantagent, Count of Anjou who is generally thought to have been the first man to bear a proper heraldic coat of arms back in 1127; his arms contained six golden lions on a blue background. However, we have no definite evidence about what arms Geoffrey's son (and Richard's father), King Henry II might have used. Usually Henry II is attributed two golden leopards on a red background. This coat of arms was certainly used by Henry II's younger son, John before he inherited the throne of England - at which point he adopted the three golden lions used by his elder brother Richard I. As King Henry II married Eleanor of Aquitaine, whose arms were a single golden lion on a red background, it has sometimes been suggested that when their son, Richard I adopted the three golden lions, he was combining the arms of his parents on one shield.
The arms of Queen Eleanor of Aquitaine
More fancifully, medieval heralds attributed two golden leopards to Henry II's great-grandfather, William the Conqueror who had died in 1087, fifty or so years before heraldry had been invented! The two lions therefore became associated with the Duchy of Normandy (in nothern France) for William the Conqueror had been Duke of Normandy before his invasion of England in 1066 led to his seizure of the English crown. It remains the coat of arms of Normandy today.
The question of whether the arms of the Dukes of Normandy, the Counts of Anjou and later, the Dukes of Aquitaine influenced the arms of England may seem academic, but it is connected to many of the reasons behind the Hundred Years War. Heraldry is often a way of visually showing the territories which a family has inherited and it was the inheritance by the English kings of these extensive lands in France which caused the tensions between the two countries.
The build up of lands in France held by English monarchs started when William the Conqueror who, as Duke of Normandy was already one of the most powerful barons in France, conquered England in 1066. From then on the Duchy of Normandy was held by members of the English royal family. The English monarchy then passed into the hands of the Counts of Anjou, so that the Anjou lands also became incorporated into the Kingdom of England. Finally, William the Conqueror's great-grandson, Henry II married Eleanor of Aquitaine, heiress to the vast duchy of Aquitaine in south-west France. Click here for the full family tree. At the height of his power, King Henry II of England actually controlled more territory in France than the French kings themselves; something which the French monarchs found very threatening.