The French and the British are always looking for an excuse to fight. In this case, it is the area south of the Great Lakes along the Ohio River. The British want control because they want to expand, but it is "French Owned" although it's technically Iroquois land. These upcoming battles will determine who gets control over North America. At the same time as all these battles, there are battles happening overseas called the 7 Years War.
The French made the first move by setting up forts in Ohio River Valley. The British sent George Washington who was only 21 at the time to command for the British. He requested the French leave their forts because the British deserved to expand their fur trade (which was now booming because of alcohol) west, but the French did not budge. The French prepared for big battle at the Ohio River but the British are smart and attack somewhere else.
The British have a navy, more numbers, and alliance with the Iroquois. The French are pretty much friends with every other First Nation group (whos combat is advanced) so the French start with an upper-hand. British move their troops into Acadia (New Brunswick and Nova Scotia) where French forces are really weak (this is where the Acadian Deportation ties in). From here, the British have access to New France's main fort, the Quebec Citadel, through the St. Lawrence.
The French have been dominating because the British have been fighting traditionally with marching and bands, while the First Nations are teaching the French Guerilla fighting styles (quick ambushing). For this reason, France gets cocky and focuses all its' troops on the 7 Years War. At one point in the war, George Washington walked away from an ambush with two bullet holes in his coat and two horses shot out from beneath him, but he didn't have a scratch.
The British tried to capture the Quebec Citadel but never could because of their fighting style, and it was crown-shaped. Then, Commander Wolfe came from overseas and got the Iroquois help. The Battle of the Plains of Abraham was the British's successful attack on the Quebec Citadel. The British had a long line of men in the field, and a group of men climbing the cliffs behind. The French came out, then the Iroquois jumped from bushes, and the climbers ran for the gate.
After that battle, the British went on to win the 7 Years War and the French and Indian war. French sign the Royal Proclamation in 1763. French gave North America to British because they were afraid "Canada" would run out of resources. For the French people living there, life was pretty much the same. 13 colonies fought for the British and gained nothing but higher taxes to repair wreckage. This caused the 13 colonies to revolt, American Revolution.