The rockets really were there, with their red glare.
In the Mysore Wars of 1780 and after, British troops were attacked with Indian rockets, developed during the reign of Tipu Sultan. Some unfired ones were brought back to the Woolwich arsenal for study, and William Congreve, ( whose father was a high official at the Royal Arsenal) became intrigued with their possibilities. With his own money, he began developing his own rockets. In 1804 he had a variety, including some quite heavy ones of 32 pounds. By the War of 1812, they were being used experimentally.
Congreve could plausibly claim that a rocket with an explosive charge could do as much damage as a cannon ball, and was far more portable than the cannon. But despite Congreve's idea of making a "rational" rocket, ( as opposed to the Indian ones assumed to be without good scientific design) it was quite difficult to make rockets that functioned consistently enough to be predictable in where they would land: in other words, they never got very accurate. So, firing rockets at Ft McHenry, a very big target, was a pretty good idea. Rockets of Congreve's type would be tried on various battlefields in the first half of the 19th c.
William Congreve : The Details of the Rocket System
To add to what u/Bodark43 has already written about Congreve rockets, rocket technology has been used on the battlefield since the late 13th century at the latest. It difficult to determine the earliest use, since the first users - the Chinese - used 火箭, "fire arrow", to refer to both incendiary arrows and rockets, but by about 1280, weapons that are clearly rockets were in use. (Similar problems result from the same word, 炮, being used for both catapults and cannon, and 槍 being used for both spear and gun (obscuring the evolution of the gun from the fire-lance).)
Early Chinese rockets, and other pre-modern rockets, used gunpowder as their propellant. To propel bullets from a gun, the gunpowder is ignited, and it produces hot gas as it burns. The pressure of the hot gas pushed the bullet out of the barrel. In a rocket, the gunpowder produces hot gas as it burns, and the pressure of the hot gas expels hot gas from the end of the rocket. Since momentum is conserved, the rocket gains momentum in the forward direction equal to the momentum of the expelled hot has in the backward direction.
Early Chinese rocket technology was diverse, with rocket-arrows (which would do damage like an ordinary arrow), incendiary rockets intended to set fire to targets, including rockets without enough thrust to get off the ground - these rockets would randomly move along the ground, and where called 地老鼠, "ground-rat" or "earth-rat". These were sometimes equipped with hooks so that they would catch on to enemy soldiers and burn them. Ground-rats were often delivered into the midst of the enemy in bombs containing 10 to over 50 of them. Similar water-skimming rockets were used, called 水鼠, "water-rat".
Chinese rocket-arrows made use of modern-style rocket nozzles, using the Venturi effect. Such rocket-arrows were in use by 1350, but may already have been in use c. 1200. Heavy rocket-arrows might weigh over 1kg, and have a range of 500m. The 14th century saw rockets with incendiary and explosive warheads, with ranges of hundreds of metres, and, remarkably, multi-stage rockets with ranges of almost 2km. These were used in naval battles as well as on land. As with the Congreve rocket, a key advantage was portability - there was no need for a heavy cannon: "one man can use it yet a hundred men will be terrified of it". There were cart-mounted and man-carried multiple-rocket launchers, winged rockets, rocket-propelled explosive torpedoes for naval warfare, and so on. The early era of Chinese gunpowder weapons was diverse, with grenades and bombs (including very large bombs thrown by catapults), a wide range of incendiary weapons (including the apparent ancestor of the gun, the fire-lance), and handguns and cannons.
While the variety of Chinese gunpowder weapons shrank as the musket and cannon came to dominate the battlefield, the rocket was adopted outside China, sometimes called the "Chinese arrow". This included India, where rockets were sometimes used in very large numbers in battles. In particular, Mysore used them with success against English forces, giving a very favourable impression of the power of rockets to the English, leading to the development of the Congreve rocket. The rockets used by Mysore used iron casings, which enabled higher internal pressures and long range (up to 2km). Old sword blades were sometimes attached to the rockets to make them more damaging to the enemy.
Gunpowder rockets continue to be used today, but as fireworks rather than weapons. Gunpowder rockets also saw practical peaceful use, for example launching signal flares (e.g., as distress signals), and to launch lines from ship to ship or from the shore for life-saving:
While the 20th century saw the development of liquid-fueled rockets, which developed into monsters capable of sending humans to the moon, solid-fueled rockets (using propellants other than gunpowder) are still around in military use - ICBMs, especially for submarine launch, are often solid-fueled for lower maintenance and quicker launch.
In addition to the response of u/Bodark43 I would like to add that what happened in early 20th century in regard to rocketry was not the invention of the rockets proper, but rather the creation of mathematical model of the atmospheric and outer space flight using the rocket engine (with Konstantin Tsiolkovsky being largely credited with the theoretical part of the endeavour with Hermann Oberth and Robert Goddard being known for their pioneer engineering work) that allowed precise calculations regarding the movement of the rocket, eventually allowing e.g. putting the objects on the orbit. But the technical idea of a rocket in general, and especially as the weapon predates the introduction of firearms.
The first surviving depictions of the rockets, in the form of an arrow attached to a small tube containing gunpowder that could have worked as a reaction engine come from the Wǔ jīng zǒng yào (武經總要) written in mid-11th century by Zheng Gongliang, although it is also argued that this projectile was a simple arrow with a firecracker acting as an additional incendiary, explosive or sonic device. Nevertheless, in the military treatise Huǒ Lóng Jīng first published in late 1390s includes depictions of *shén huǒ jiàn pái (*神火箭 牌), or a ground battery for the arrows propelled by a rocket engine, arguably the precursor of the singijeon rockets and hwacha battery common in late 15th century Korea (the name of the latter seems to be borrowed directly from the Chinese huǒ chē, lit. 'fire cart'). What is interesting, that the artillery manual Artis Magnae Artilleriae pars prima written by a Polish artillery officer Kazimierz Siemienowicz and published in 1650 in Netherlands, although focusing on the classical artillery, includes many concepts related to rocketry, including multi-stage designs and fin-stabilized projectiles, that won't be introduced until 20th century. The rocket became widespread, if still not popular in European and American armies throughout the 19th century, especially after the adoption of the steel-body rockets by British who first met such weapon in Mysore ruled by an already mentioned Tipu Sultan (the idea of using iron-cased rockets is attributed to his father, Hyder Ali Khan).
Thanks for the insight guys and gals!
