Outcome: Freedom for the United States; Americans fail to annex Canada
Most of Canada was French territory until the end of the French and Indian War. When cries for independence emerged from the thirteen colonies, French and loyalist Canada favored the King. General George Washington favored a military campaign to annex the region. An invasion spearheaded by Richard Montgomery and future traitor Benedict Arnold quickly took Montreal but was defeated outside Quebec City. After the failed invasion the United States realized the impracticability of taking Canada. However, the United States Congress pre-approved Canada's membership in the Articles of Confederation; Canada never took up the offer.
War of 1812
Outcome: United States fails to take Canada, British temporally take parts of the United States
Americans like to sway they "beat the British twice." Not quite. The invasion of Canada proved to be a disaster. The fighting started off well but most of America's army was militia and unwilling to fight off their home state's soil. The highlight of the American invasion was the burning of York (present-day Toronto) but that led to the British burning of Washington D.C. At the end of the war the British controlled Fort Detroit (Detroit, Michigan) and Fort Niagara (Niagara Falls, New York).
Lower Canada Rebellion, Patriot War, and the Republic of Canada
Outcome: Defeat of the pro-American Canadians by pro-British Canadians
Canada had a series of rebellions which were crushed. In the 1830s, the Pariote Movement of French and Anglo-Canadians favored liberalism and an independent Canada. They were backed by the Hunter Brothers, a secret society dedicated to the liberation of Canada. The Lower Canada Rebellion and the Patriot War were a series of wars in which the British solidified their presences in Canada. Some Hunter Brothers were Americans and were led into battle by Americans. The rebellions mostly ended when the Republic of Canada's government was forced out of the Niagara River and into American jails.
Outcome: Canada takes part of Maine
Look at old maps of Maine and one may notice that Maine had much more territory up north. That was the claim laid by the Americans ever since the Treaty of Paris. The United Kingdom; however, claimed much of Maine. The Aroostook War settled the matter by negotiation. Competing maps, both showing the holder having less territory, pressed each side into compromise.
San Juan Island War
Outcome: United States wins one; keeps several small islands in present-day Washington state
The San Juan Island War is also referred to as the Pig War. An American killed a pig in his garden and all Hell nearly breaks loose. The pig belong to an British subject on the disputed island. Anger rose on both sides because the islands were strong points along valued trade routes. Both sides agreed to joint military occupation after a rushing of troops including future Civil War (non)-hero George Pickett. Kaiser Wilhelm I eventually was asked to settle the matter. He ruled in favor of the United States.
Outcome: British and Canadians withstand Irish-American attacks
The United States was upset at the British tact support for the Confederate States of America during the Civil War. There was one way the Americans could strike back without risk of major backlash: terrorism. The Fenian Brotherhood was an Irish republican/terrorist organization based in the United States. A Brotherhood song ending with "And we'll go and capture Canada, for we've nothing else to do" sums up the mood of Irish who just wanted to kill some British. It is alleged President Andrew Johnson gave passive support to the Brotherhood though American government officials stopped some raids before they occured. A series of raids and battles, some won by the Brotherhood, failed to do anything except make Canadian/American relations distant until World War II.
The "wars" between United States and Canada is surprisingly one of Canadian success. However, World War II required cooperation and brought America and Canada closer together. Today, Americans and Canadians regard each other as brothers, if somewhat odd brothers.