Map of San Juan Islands, showing the possible channels which the border could follow. Blue: Haro Strait, green: San Juan Channel, red: Rosario Strait
The Pig War was a confrontation between the USA and the British, who had control over Canada at the time. It's a dispute about the San Juan Islands, as a result of the Treaty of Oregon, which stated that the border of the USA and Canada would follow the middle of the channel between the USA and Vancouver Island.
The Treaty of Oregon was too vague, since there are three possible channels the treaty could have reffered to. Both parties strongly believed that the islands were theirs.
It's called the Pig War because the war almost really started because of the killing of a pig. Lyman Cutler settled on San Juan Islands, along with some other Americans. He had a small potato farm, but only fenced it from three sides. This led to a pig, owned by Charles Griffin (farmer), to eat some of his potatoes, eventually resulting in Cutler shooting the animal.
The dispute escalated because of general William S. Harney ordering captain George Pickett to place troops on San Juan Island. In response to that, the British ordered the Royal Navy to keep an eye on the Americans on the disputed island. James Douglas (governor) ordered admiral Geoffrey Hornby to also land on the islands, which caused the Americans to be ordered to shoot the British if the they did so.
In the end, they let the Kaiser Wilhelm I of Germany decide. The kaiser ruled in favor of the USA, which means the San Juan Islands are now officially US territory. This 'war' is iconic, since the only casualty in the entire war is a pig.