The most prolific type of relationship between native groups and the British (or Americans or French or Spanish or any other regional European power) was one of trade. Trade was much more multifaceted than strictly exchanging goods; it was a measure of relationships, of social esteem and political power, it fostered and maintained alliances, created kinship ties, broke down language barriers, and generally allowed each different group to get to know, and trust, the other.
By the War of 1812, the relationship between the British and the Great Lakes native nations was complicated. A couple of decades before, during and after the "Northwest Indian War" waged by the United States against the Wabash Confederacy (a loose coalition of Great Lakes and midwest groups headed by the Shawnee), the British had promised material and political support for the Confederacy against the US, but reneged on their promises when it looked likely that a shooting war might develop between Britain and the US. This fractured the alliances that had formed in part due to the War for Independence, when again the vast majority of Old Northwest groups allied with the British against the United States.
This fracture had never quite healed, but relationships of trade and diplomacy remained. On the US side, Detroit and Mackinac were hubs of diplomatic efforts, trade, councils, and treaty annuity distribution, and forts Malden and St. Joseph were the British counterparts. Due to the often convoluted history of formal and informal agreements between the two powers, payments of goods, tools, weapons (and their subsequent maintenance), and cash were distributed through Indian Agents at these locations, and it was quite common for some tribes to dip in and out of both powers' pockets.
They managed this mostly because British support, even though it was often tepid and ineffectual, meant that there was no particular monopoly of political and military power in the region. Natives could play both groups against the other, promising support or trade or kinship to one while threatening that if their desires weren't met, they would take their business elsewhere.
A ton of importance rested on the government representatives for the US and Britain in the form of Indian Agents. It was their force of personality that could make a difference between firm support or empty promises, and it was their ability to persuade, cajole, promise and support native groups that would mean the difference between a group of native warriors staying at home when the shooting started, or showing up and fighting.
These relationships were necessarily far more informal than that of, say, John Quincy Adams and the Russian government. Power among native groups was far more distributed and far more fluid than European governments, which meant that agreements on paper would not often be maintained as consistently, and this meant that the diplomatic relationship between regional European powers and the native powers was hard to assess.
So by the time the War of 1812 started, no one was quite sure of the amount and quality of support that the natives would extend to the British. Though the historiography often represents the Indian choice as an easy one - the British were fairly objectively less intrusive and land-hungry than the Americans - it was left to individual chiefs and prominent warriors and leaders to choose their relationship, or to stay neutral.
A few hundred natives made a massive difference in the early part of the War of 1812, with victories at Mackinac, Detroit, Fort Dearborn, and the River Raisin accomplished mostly because of efforts of Native support, which demonstrates how crucial the relationship could be.
The rest of the war is beyond the scope of this (already too long) post, but suffice to say that Native support continued with some enthusiasm for the duration of the war, despite losses of prominent leaders like Tecumseh.
In the end, the British pulled support for Native efforts off the diplomatic table to end the war. It had been a sticking point for the American diplomatic efforts, and the British essentially felt that it was too much trouble, and supporting Natives was too likely to pull the British into a war they didn't want to fight. Early promises of support for a neutral Native nation in the Great Lakes region, for political and military support after the war, dried up, and the British no longer interfered in the relationship between the US and the Native population afterward.
The War of 1812 broke the push-and-pull political dynamic that had dominated Indian relationships with Europeans since the first establishment of colonies. It meant that the sole power in determining Native relationships was the United States, and this immediately manifested in a much more uneven power dynamic than had been there before. The increasing monopolization of the fur trade by American companies in the 1820s and 30s and the eventual collapse of the trade as a meaningful political bargaining chip among Native groups eroded their power base and eased the way toward the removal policies that most American presidents favored.
There was also a change from dealing with chiefs as representatives of certain groups to dealing with American-defined "nations." For instance, the Odawa people had, in treaties prior to the War of 1812, been referred to by their group names in general and to the particular individuals who signed agreements in specific, to a more general reference to "The Ottawa Nation." Other tribal groups and bands were similarly clustered into "nations" that the US dealt with diplomatically. Though prominent individuals still played a role in the diplomatic relationship with the United States, the key feature was that any agreement made by any chief related to the "Ottawa Nation" was considered a binding contract for all of them. It was a subtle change, but an important one. Imagine a changing relationship at a workplace where you had gone from a part owner to a holder of stock, for instance: in many ways the same, in other ways your power was vastly reduced.
One point I'd like to make before I stop rambling: it is common to talk of Natives as being under "control" of European powers, but I would argue that, especially prior to the War of 1812, there was a much more give-and-take relationship and interdependence between Native powers and European ones. Richard White's "Middle Ground" argues this exceptionally well - European power could not be manifested as powerfully on the periphery of their influence as it could at the core, and this played into the complicated reputation networks of Native groups in a way that further diffused European power. The Great Lakes region was one of the areas where this manifested most intensely.
Obviously Richard White's The Middle Ground is where you want to start reading. For a more general history of Native Americcan politics and economy, check out Facing East from Indian Country by Daniel Richter.
Much more about the War of 1812 I'd say the best single work is John Sugden's Tecumseh's Last Stand. it focuses more on Tecumseh and the Thames campaign than anything else, but has a great survey of Indian diplomacy and military support for the British throughout the war.
I am of course open for follow-up questions, if you have them.