The War Hawks were a group of Republican Congressmen who, at the end of the first decade of the 1800s, demanded that the United States declare war against Great Britain, invade British Canada, and expel the Spanish from Florida.
At the time, ongoing conflicts between Great Britain and France had disrupted trade between the United States and Europe, leading to some hard economic times for American merchants. At the same time, British Canada was giving secret (and sometimes not-so-secret) aid to the Native American tribes that were hindering U.S. expansion in the west.
President Thomas Jefferson and his successor, James Madison, both tried to remain neutral in the wars between France and Great Britain, using diplomacy and economic pressures to keep America's European trade routes flowing. In the end, though, most of their policies failed.
In June of 1812, the War Hawks got what they wanted: Congress declared war against Great Britain, thus beginning what would come to be known as the War of 1812.
The name War Hawks was created informally by Congressman John Randolph, who was opposed to entry into the war. Because it wasn't an official designation, there was never a roster of who was and who wasn't a War Hawk, although Speaker of the House Henry Clay and South Carolina Representative John C. Calhoun (who would go on to become Vice President) were certainly among their numbers.