This is a pretty quick explanation. Let's say country A and B are at war and country C, for a completely unrelated reason to the original war, declares war on country A. Country C is now a co-belligerent in the war between country A and B. B & C are fighting a common enemy without an alliance or treaty binding them to attack together. They're fighting a common foe for different reasons.

That's precisely what happened in this case. The United Kingdom would be taking part as a belligerent opposed to France in the War of the 6th Coalition -- which would take place between 1812 and 1814. The United States would, at the same time and unrelated with an alliance with France, declare war on the United Kingdom over rising tensions related to impressment (forced service) of American sailors and supporting/funding anti-American Native Americans which were impeding Westward expansion.

So the United States and United Kingdom would wage war against each other at the same time the United Kingdom would be fighting France without an alliance between France and the U.S. That would make the U.S. a co-belligerent in the 6th Coalition War.