The War of 1812 ended with the signing of the Treaty of Ghent in 1814 and when the diplomats finished carving up the boundaries between the United States and British North America, today's Canada, not much had changed hands.
Certainly the boundary line along the Great Lakes remained the same, as well as along the 45th parallel dividing Quebec from New York and Vermont. There was much yet to be discussed with the Maine border between Quebec and New Brunswick, but this was established with the Webster/Ashburton settlement of 1842.
Across the Prairies the Treaty of 1818 establishing the 49th parallel as the boundary from the Lake of the Woods west to the summit of the Rocky Mountains. This gave the Americans the upper Red River Valley in today's North Dakota and Minnesota. While further west the very southern tips of Saskatchewan and Alberta have a small section of the the Missouri-Mississippi rivers basin.
But west of the Rocky Mountains the area from the 42° to 54° 40' North remained under joint occupation of the British and American governments. A situation that was not satisfactory with either London or Washington.
The issue continued to fester away. At first it was not important for either nation. The Americans were busy dealing with the boundary issues between them and the Mexicans. For the British this part of the Pacific Northwest was not an official part of their empire as it was still under Hudson's Bay Company control and was just seen as a mercantile operation that did not have value beyond the fur trade.
Through the 1830s and 40s this began to change as American settlers moved westward to the Willamette Valley in today's State of Oregon. Gradually the Hudson's Bay Company relinquished control. The settlers were drawn to the fertile regions south of the Columbia River and in first step the British gave up what is today Oregon.
Certainly American politics refocused on the question of the Northwest problem. During the 1844 presidential campaign Democratic presidential candidate James K. Polk ran on a platform of taking control over the entire Oregon Territory.
When Polk became president a more sobering approach was taken and the discussions between Washington and London focused on where to draw the line between the two countries. After many negotiations the concept of just drawing the border west from the summit of Rockies to the middle of the Strait of Georgia was decided on. As easy as this as this was during the discussion it lead to three key issues that have been a part of the Washington and British Columbia history.
1. By defining the middle of the Strait of Georgia it created the American enclave of Point Roberts, which is a part of the mainland that is below the 49th parallel, but bounded on three sides by water. This leaves the only land access through Canada.
2. It created the situation of which nation was responsible for the San Juan Islands. Which lead to the Pig War from 1859 to 1874. The only casulity was a pig, but the San Juans were in dispute during this period.
3. As many of the mountain ranges and valleys of British Columbia run north to south ending deep into Washington state, it led to many transportation and commerce routes beginning in the States to serve Canadian resources. The best example is Spokane, Washington which grew prosperous on gold and silver being mined in to British Columbia's Kootenay region.
RESOURCES:
US Department of State, Office of the Historian: http://history.state.gov/milestones/1830-1860/OregonTerritory
L’Encyclopédie de l’histoire du Québec / The Quebec History Encyclopedia: http://history.state.gov/milestones/1830-1860/OregonTerritory