Trade and political divisions.
In the early years of the Republic, the United States had two major political factions: the Federalists and the Republicans. (It should be noted that the Republicans share little more than a name with today's Republican Party.) The Federalists traditionally favored an aristocratic ideal of government, with a small, elite corps of politicians making saavy, disinterested decisions on behalf of their constituents. The Republicans took a populist view and believed the people, no matter how uneducated or low-class, were the folks best able to look out for their own interests.
Since 1793, the great choice in the Western world was whether to support Great Britain or whether to support France in the wars sweeping Europe. The Republicans, as you might expect from their beliefs, strongly supported Revolutionary France. The Federalists favored Great Britain, seeing it as a parent to the American republic.
Despite these political divisions, the United States pursued a policy of neutrality throughout most of the Napoleonic Wars because of the belief that peace was good for business. Americans at the time firmly believed that ties of commerce, not diplomacy or warfare, were the best way for the United States to prosper. For the most part, they were right.
During the Napoleonic Wars, America's shipbuilding and oceangoing trade boomed enormously. Already by 1795, the United States' economy was four times larger than it had been before the Revolution. This increase accelerated despite intermittent embargoes against the British and French and the occasional policies of seizure by both combatants. The United States kept its neutrality even when HMS Leopard attacked USS Chesapeake, a U.S. Navy warship, in 1807.
By 1812, however, the United States was becoming fed up with the policy of impressment conducted by the Royal Navy. Under this policy, the British Navy boarded any and all ships in international waters, seeking British seamen who had enlisted in foreign militaries or were serving aboard foreign merchantmen. Given America's liberal immigration policies and a common language with Great Britain, Americans were seized incidentally.
In the American West, settlers were convinced that Great Britain was arming American Indians in combat against them, despite treaties to the contrary. New states like Tennessee and Kentucky sent aggressive delegates to Congress who called for war against the British.
In 1812, the vote to declare war was 79-49 in the House and 19-13 in the Senate, the closest vote to declare war ever cast by Congress. The vote was tightly along party lines, with every "yes" vote coming from a Republican Congressman. Not a single Federalist voted for war.
In 1812, Federalists were concentrated in New England. One might think that the issue of impressment would be high on the priority list of these gentlemen. It was, but the disruption of trade and partisan divides were higher still. The outbreak of war meant American shipping was now a target of the Royal Navy. America lost its No. 1 trading partner, Great Britain, and new taxes were levied on imports to pay for the war. All of these things were anathema to Federalists.
Republicans, on the other hand, were enthusiastic about the war. They believed that combat was necessary to protect the West, to fight for individual liberty in Europe, and to protect the sovereignty of the United States against British impressment. While the coast of the mid-Atlantic and South also depended upon the maritime trade, it did so to a much lower degree than New England. Its priorities were different, and these regions supported the war.
/u/The_Alaskan has given an excellent answer, but I want to expand on some points, as well. The causes of the War of 1812 expose some very interesting political dynamics, many of which have to do with policies of the Jefferson administration.
It's axiomatic now to think of Jefferson as one of the US's greatest presidents, but it's important to remember that he was elected in a deeply divisive time and had deeply divisive notions about how the country should go forward. The Louisiana Purchase, for instance, was enormously expensive, and to many Federalist minds, it would lead to a serious imbalance in the power dynamics of the country. Basically, huge land acquisitions, unless they were carefully (and centrally) managed, would lead to wealthy men purchasing cheap land, only to sell it at a much higher profit to other settlers, which would naturally result in a small group of wealthy country elites. Additionally, Federalists - especially during the Washington and Adams administrations, felt that the number 1 reason for Indian conflicts was the sudden influx of treaty-breaking settlers. The French had an entirely different colonial paradigm, and Americans and their presence tended to aggravate border issues that inevitably led to costly conflict. There were other issues, but for the sake of brevity, Federalists believed that the Louisiana Purchase would threaten the unity of the nation because of a sudden shift in power dynamics and the threat of conflict with Native Americans. It is also worth considering that Jefferson was loathe to pay for anything at all, and when he suddenly found an obscene amount of money to pay for a massive land expansion it was viewed suspiciously.
Another policy, one which often goes entirely unremarked, is actual a series of policies collectively referred to as the Restrictive Acts. Following the Chesapeake Affair - which is summarized in /u/The_Alaskan's post - Jefferson sought to censure Great Britain by curtailing American trade. It eventually basically outlawed overseas trade entirely. This can be viewed as a policy that, in many ways, echoes the popular boycotts of the revolutionary period, and was intended - if we are to believe Jefferson - to hit Great Britain in the pocketbook by eliminating American goods from the British trade portfolio. It could also have been viewed - as it was by many Federalists - as a gross and pointed attack on Federalists themselves. Remember that, concentrated in the North Eastern states, the Federalist party had a number of merchants, ship builders, insurance agents, bankers, and other financial elites in their ranks, and restricting trade hit them extremely hard. It led to a regional depression and widespread unemployment in the North East, and stoked further animosity between the Federalists and the Jefferson administration.
In the end, the Restrictive Acts were a complete failure. It essentially bottomed out the American economy and barely dented Great Britain's economy at all. It was also impossible to enforce, and many in the border states openly traded with Canada (which went on even into the War of 1812). The acts were revoked by the end of Jefferson's administration.
Federalists had many, many reasons to suspect that Jefferson was attempting to undermine them, and to exert a greater influence of "aristocratic democrats" in US government (the term "democrat" has an entirely different connotation today than it did in 1804, by the way, and was essentially shorthand for "mob rule," which was one of the many criticisms leveled that the Jeffersonian style of government. A little more on that later). Timothy Pickering in 1804 floated the idea of a Northern Confederacy, which would remove New England from the control and domination of the aristocratic and slave-owning south. There were many Federalists at this time who felt that the Constitution had given away too many rights to slave owners, and that the entire document was compromised (quite a few modern historians make a much bigger deal about Federalist abolitionists than I will here, suffice to say that there was a small and vocal abolitionist minority within the party). Pickering's idea was basically to sever the economic and social ties to the south - as many already believed that the US was made of two entirely irreconcilable countries parading as a united political entity. In any case, Pickering's effort to create a separate New England failed, and subsequently, the Federalists were increasingly looked upon as an out-of-touch and diminishing regional power and Pickering's efforts were mocked as a reactionary conspiracy.
The War of 1812 changed that. Jefferson's policies were believed by many to have been a major cause of the war, as well as the rising influence of the western states, many of whom felt that in order to sever the British influence to the North American Indians and to protect the US from further British meddling, the US needed to capture Canada. Henry Clay was a major influence in calling for an invasion of Canada, and Jefferson himself was, if not entirely in support of the measure, confident of the US's ability to easily swoop in and capture the country.
You can see how the threatened invasion of Canada pushed a lot of Federalist warning buttons. Federalists had always been careful of war and hoped to avoid it, and had even laid out an impressive roadmap for international conflict - especially as regards seafaring problems - in the Quasi War with France, and advocated a limited naval engagement against Great Britain or even an alliance with Britain against Napoleon. Additionally, as the call for an invasion proved enormously popular, it again called to mind the ruinous clamor of "mob rule" that had turned the French Revolution into a bloody nightmare.
This was further proven during the Baltimore Riots, a period of about a month of continued violent rioting shortly after war was declared that initially targeted a Federalist newspaper press, then went on to target Federalists themselves, then minorities and immigrants. This was mob rule at its absolute worst, and the open animosity toward Federalists and those who were against the war led many to believe that the mobs were actually being directed by the Madison administration to silence the anti-war press.
So how does this all lead to the Hartford Convention? A decade-long period of diminishing power of the Federalist party, of open federal attack on their livelihood and way of life, and an increase in mob violence, a clamor for aggressive warfare that would further unbalance the power dynamic in the country, not to mention the very real possibility that Great Britain would absolutely dominate the war from the onset. Federalists had many reasons to consider secession and a separate peace with Great Britain. Ultimately however, the Hartford Convention was a discussion and a threat, and many historians believe that the majority of Federalists were not even close to being in favor of a secession at all, but felt that the threat of it was enough to send a wake-up call to President Madison.
As it stood, the unlikely and self-declared "victory" in the war cemented Jeffersonian policies and set the paradigm for next decade, which was dominated by Jeffersonian policies and ideals. Federalist ideals never entirely went away, but the party itself was a smoking ruin by the 1820s.
Sources: "Escaping Mr Jefferson's Plan of Descruction" by Kevin Gannon - available on JSTOR if you have access
Mr and Mrs Madison's War by Hugh Howard - lots of info about Madison's reaction to the Hartford Convention
"The Baltimore Riots and the Breakdown of the Anglo-American Mob Tradition" by Paul Gilje (also on JSTOR)
The Civil War of 1812 by Alan Taylor
Eagle and Sword: The Federalists and the Creation of the Military Establishment in America, 1783-1802 By Richard Kohn. It's an earlier period, but it gives a lot of information about Federalist military policies and their antithesis in the early Jefferson administration
The War of 1812: A Forgotten Conflict by Donald Hickey. It's a general history and can be quite biased, but it gives a great overview of the anti-war efforts of the Federalist party throughout the war.