The rockets used by the British during the War of 1812 were the work of William Congreve Jr., who was inspired by rockets used in India against the British. Using political influence through his father, William Congreve Sr., he was able to convince William Pitt and John Pitt, master general of ordnance, to produce rockets on a large scale for the wars against Napoleon.

You might look up Simon Werrett's paper "William Congreve's Rational Rockets" in the March 20, 2009 issue of Notes and Records of the Royal Society of London for a more detailed description of Congreve Jr.'s efforts. Frank Winter's The First Golden Age of Rocketry: Congreve and Hale Rockets of the Nineteenth Century is a published book that focuses on these rockets specifically.

In short, he promoted his rockets as a means of providing artillery cheaply, without the extensive use of hired labor that the Royal Artillery then needed for hauling artillery and producing ammunition. In 1807, Congreve explained that "It is ammunition without ordnance, it is the soul of artillery without the body."

Congreve's rockets were produced in 10 calibers and used both incendiary and explosive warheads, but they generally looked not unlike your common can or bottle rockets for fireworks today. Instead of stabilizing fins, they were balanced through the use of a long wooden stick that extended downward from the warhead, which was conical or hemispherical. The wooden brace was used both as a firing aid and to improve accuracy, but accuracy was never good, and the explosive impact of the rockets was minimal.

Congreve, however, had the connections to make even failure look good. His rockets were used against the French port of Boulogne, in the bombardment of Copenhagen, in Baltimore, at Leipzig, on the Barbary Coast of Africa, and in India, as well as at the Battle of New Orleans.

At Boulogne, Sir William Sydney Smith oversaw the firing of nearly 600 rockets. Viscount Keith, commander of the North Sea Fleet, was a skeptic, and from the limited results at Boulogne, he saw that "Congreve rockets could not reach by miles ... the rockets were fired without effect, some of them burst in our own boats, and none went in the intended direction."

Smith, an apologist for the rockets, called them "a valuable addition to the force of swift-rowing boats, as the smallest can throw a six-pounder [warhead] ... without being encumbered with the weight of ordnance."

But even he had to admit that using the rockets could be as dangerous to the people firing them as those receiving them. "This must be carefully kept and skillfully managed, or there is danger in the application," he wrote after the fact.

Rockets were used in the Chesapeake campaign, as illuminated by the Star Spangled Banner, but even when fired from land instead of sea, they were inaccurate and lacked the force of massed guns. Their principal effect was to induce terror and erode morale on the American side, and in that, they were effective. At the Battle of Bladensberg, American militia fled under the sound and impact of rocket fire, allowing the British forces to effectively turn the flank of the Americans and complete the capture of Washington, D.C.

Against a properly prepared opponent, particularly one dug in, the effect was limited. In Baltimore, the effect was similar to what was seen at Boulogne. The ship carrying rockets, the Erebus, was outranged by the guns of Fort McHenry, and it couldn't close to effective rocket range. Most of the rockets it fired were undershots that missed the fort entirely.

A few years later, the British employed the Congreve against the defenders of New Orleans. Major Arsène Lacarrière Latour, who wrote a memoir of his time in West Florida and Louisiana, described what it was like to fight under their fire.

"During the whole day, the enemy incessantly threw Congreve rockets, which wounded some of our men. By one of these, Major Carmick, of the marines, had his horse killed, and was himself wounded in the hand. The British had great expectation from the effect of this weapon, against an enemy who had never seen it before. They hoped that its very noise would strike terror into us; but we soon grew accustomed to it, and thought it little formidable; for in the whole course of the campaign, the rockets only wounded ten men, and blew up two caissons. That weapon must doubtless be effectual to throw amongst squadrons of cavalry, and frighten the horses, or to set fire to houses; but from the impossibility of directing it with any certainty, it will ever bee a very precarious weapon to use against troops drawn up in line of battle, or behind ramparts."

A British perspective on New Orleans can be found in George Gleig's 1825 memoir. In that fascinating book (it's still good reading!) he writes of one point in the fighting that "a few rockets were discharged, which made a beautiful appearance in the air; but the rocket is at best an uncertain weapon and these deviated too far from their object to produce even terror amongst those against whom they were directed."

A.D. McCaig's 2000 paper entitled "The Soul of Artillery: Congreve's Rockets and their Effectiveness in Warfare" and published in Journal of the Society for Army Historical Research seems to be exactly what you're looking for, but I can't find it digitized freely, unfortunately.

The rockets used in the bombardment of Fort McHenry were called "Congreve Rockets" after their designer, Sir William Congreve. They were developed in 1804 and were based on rockets that had been used against the British in their wars against the Mysore Kingdom in India.

In appearance, they can generally be said to resemble a giant version of a modern day bottle rocket. There were different types, but most were tipped with explosive artillery shells.

In size, there were 3 categories: Heavy, Medium, and Light. They ranged in size from 6 feet long with a 27 foot stick and weighing 300 pounds, down to 6 pounders, 2 feet long with an 8 foot stick. The particular rockets used at Fort McHenry were 32 pounders fired from the HMS Erebus

The rockets would be set into tubes angled in the direction of the enemy. Once lit, there was no way to control where the rocket went. They were as such generally only effective as a way to bombard a large area, such as a fort as in this example. What they lacked in accuracy they partially made up for in a relatively fast rate of fire, and ease of travel and deployment as opposed to regular muzzle loaded artillery of the time.

These rockets were used by the British in the Napoleonic Wars, and continued to be in use in combat by them up to the 1850s, when a new rocket designed by William Hale that was much more accurate was introduced.

Two comments here have talked about Congreve rockets' battlefield effectiveness. What about the mortars that are the other part of the line?