2 posts coming up: (1) Orbits and Mathematics (for orbit, brightness, tails); (2) Design considerations
The dust tail (yellow, reflecting sunlight) and the ion tail (blue, fluorescence).
In the beginning, the Solar System was just dust and water rotating about the Sun. Dust clumped together, water clumped together, and planets were formed. The closest-to-the-Sun planets are largely made out of rock because the heat of the Sun didn't let ice stay around, while the rest of the planets are giant bodies of gas and ice. Comets are the latter planets' left overs. They're known as dirty snowballs because they're largely made of ice, carbon monoxide, and other gases like argon, ammonia, and such. This is contained in a dark crust of complex organic compounds, which absorbs light. Comet nuclei are actually some of the darkest observable material. We don't easily observe this because when the Sun isn't close enough to light up the comet, the nucleus is too small. When the Sun is, the nucleus will be obscured by a large coma. There is an even larger cloud of hydrogen, but this is not visible to the human eye.
There are thousands of comets hypothesized to exist in Oort's Cloud in the outer solar system and the Kuiper Belt, which rotate Oort's Cloud accounts for long period comets (> 200 years) and the comets can stay there, undetected, and remain stable far from the Sun. Kupier's Belt is nearer and accounts for short period comets (<= 200). The comets lay dormant in the cloud, but due the gravitational pull from the planets, its orbit became highly eccentric- an elongated ellipse.Trom the highly eccentric orbits of the comets, astronomers have extrapolated their existence. Jupiter and other massive bodies gravitational perturb the comets, comets collide with comets, and so, in this way, comets leave the Cloud or Belt.
As the comets near the Sun, they receive enough heat energy (especially with their dark crusts) to start outgassing. In other words, the frozen H20 and gases sublime from solid to gas, not via water since Space is a vacuum, and spew out of the comet. Sunlight and the photoelectric effect ionizes water molecules into Hydrogen ions and Oxygen ions. Meanwhile, a great deal of solar wind is coming at the comet. Solar winds are ionized particles from the Sun moving at such high speeds that they escape Sun's gravity. Since the comet is traveling at supersonic speeds relative to the solar wind, a bow shock is formed. In other words, the high solar wind speed dips drastically and ions from the Sun accumulate because they arrive faster at the bow shock than they can leave. A magnetic field (magnetosphere) flows around the comet like a stream around a rock, and the ionized plasma from the comet will form the ion tail. This ion tail is long and narrow, and will point directly away from the Sun because of the magnetic effect. The ion tail is actually made of plasma, and is blue because of CO+.
There was dust, mainly silicate, mixed into the gas and water, and they also spew out of the comet. This forms the coma, which can hide the nucleus. Radiation pressure bounces sunlight off the dust particles, creating the dust tail. It reflects a lot of light, making it shiny therefore observable. The dust tail doesn't directly extend away from the Sun like the ion tail, but curves. This is because the nucleus rotates, and so sunlight doesn't hit the nucleus uniformly: indirect sunlight, less outgassing; direct sunlight, more outgassing. Thus the tail curves along the trajectory. Depending on the nucleus' rotation, the comet can either be propelled or repelled, affecting its orbital radius. That's why Comet Encke's period decreases.
There can be other hard-to-detect tails, such as the faint 50Mkm long Sodium (Na) tail of Hale-Bopp's. Sometimes, a comet will have multiple ion tails (Comet Donati), or when it breaks apart it can have a fan of tails (Comet West).
All this material-spewing occurs at "vents", or puncture points in the comet's crust that cracked under the pressure of the volatile material wanting to get out. We prove the existence of vents by studying the tail. In Hale-Bopp, the moving tail showed that the comet was rotating because depending on which side was facing the Sun, the volatile materials would spew out from the vents on that side.
Now, for the composition of the comets. It was once hypothesized that Earth's water comes from comet collisions, since it was suspected that the Deuterium (isotope of H) to Hydrogen was the same in both bodies. Comets also reveal a lot about the Early Universe. The above theory, however, is dubious because it was later confirmed via samples that the ratio is different. A comet's chemistry can affect their color (the gas may fluorescence under UV rays). For example, yellow Sodium tails and Argon fluorescence. The main thing is that Comets is made of (1) crust: complex organic compounds, silicates (metal like Iron and Sodium, silicon to form rock); (2) volatiles: H20, CO mainly, then Na, CO2, OH, NH4, HCN, CH4, H2S, Ar, etc.
