It has multiple layers to it, both as a conflict between America and Britain and an internal conflict between New England and the Southern US.
It came about when it did as a move of opportunism. There were multiple factions in America that wanted different things and the war was highly controversial. New England was opposed to the war, traded with Britain during it (it wasn't included in the British blockade at first), didn't send troops to invade Canada in the initial wave and even entertained the idea of secession.
The ''warhawks'', which included pro-war American politicians such as Henry Clay and John Calhoun, wanted to invade Canada and sold the war as being a conflict waged to assert independence and end impressment, but this was just a propaganda move more than anything.
Impressment was a layered issue in itself with many, but obviously not all, sailors being British deserters who went to America with the aim of escaping the horror of the Napoleonic Wars. Obviously, American sailors were pressed, but this wasn't a real war aim and the practice ended when Napoleon was defeated in 1814.
The US attacked when it did because Britain was fighting Napoleon and the conquest of Canada was a real, though not a universal, war aim. Different politicians in America wanted different things, some did indeed want to conquer land, others wanted to just fight. In any case, it wasn't a ''Second War of Independence'' or a continuation of the Revolutionary War by any means.
For Britain, the war was a sideshow that they just wanted to end as soon as they could and focus on fighting Napoleon. They wanted to defend Canada and preserve the status quo. Later on in the war, some voices in Britain wanted to launch an invasion of the US and maybe take territory, but this was never an official policy.
There is also an issue of native tribes. Americans started expanding west into the area of the Great Lakes, Florida and the area of the Louisiana Purchase as early as the 1790s and early 1800s, and consequently came into conflicts with the Native Americans. The Great Lakes tribes are the most relevant here, as Britain supported them in their fights against Americans, traded with them and supplied them with weapons. This was another issue for the Americans, but the war against the natives started against Tecumseh in 1810 and ended in 1813, with Tecumseh's death in the Battle of the Thames. I see this as a separate conflict that later overlapped with the War of 1812, but was ultimately a different theatre than the main war against Britain itself.
It gets even more complicated after the war ends, because its effects impacted both the Canadian and American national psyche for the next several decades.
The US failed their objectives and couldn't really hold any claims of a military victory, it ended up nearly bankrupt, blockaded, its military was expelled from Canada, its economy sinking, Washington was set on fire, the coast was being constantly raided by the British and the political situation looked grim. The only silver lining consisted of a few victories in raids such as Baltimore or Plattsburgh. After the war ended, Madison's government initiated a propaganda campaign that presented the war as some great victory where the young nation of Americans held their own against the mighty British Empire and was able to stand up to their oppressors. This view, as has been shown, isn't accurate at all and the US was largely defeated in military and economic terms, their only claim to success being that they weren't completely overrun, which the British didn't even attempt to do anyway. Nevertheless, this myth of a ''Second War of Independence'' was sold to the public and they believed they won the war.
Britain wanted to end the war, as has been said, and didn't really pay it much mind anyway, being more concerned with Napoleon and later the Congress of Vienna.
For Canada, this war presented one of the first instances of Canadian nationalism and made Canadians feel like they were indeed separate from Britain and were strong enough to defend themselves against American invaders. The Canadian defense in the war served as a springboard for a separate Canadian identity and played an important part in Canada's own emerging national myths. The Canadians, or the colonists that would become Canadians in a generation, were the biggest winners in this war for this standpoint.
The biggest losers were the Native Americans, who had their independence crushed and were further integrated into the US.
A very complex conflict with many sides to it, with very confusing war goals on both sides that evolved as the war went on.
