Chi Cygni is a red variable star at a distance of 520 light years (160 pc) [1,2] in the constellation Cygnus. If you want to find it in the sky, it's located within the longer part of the cross that forms the center of Cygnus. The binary star system at the southwestern tip of the cross is named Albireo, while the bright star halfway between Alberio and the center of the cross is named Eta Cygni. Chi Cygni is located just southwest of Eta Cygni. However, like I said, it's a variable star, so depending on when you look, you may need a telescope.
Chi Cygni is one of many stars called Mira variables, and they are named after the star Mira in the constellation Cetus (which is a much lamer constellation than the constellation Cygnus). Originally these stars were non-variable stars about the same size as the Sun (or, to be more precise, with masses 8 times the mass of the Sun or less) that fused hydrogen into helium in their cores [3]. However, their cores eventually filled up with helium, and after this, the stars became red giants and went through a phase where that helium in the stars' cores was fused into carbon or oxygen. Now, the cores of Mira variables are completely filled with carbon and oxygen, and because the stars aren't that large, they don't have the mass to trigger the fusion of that carbon and oxygen into heavier elements, so the carbon and oxygen just sits there [3]. However, the fusion of hydrogen into helium continue is shells around the cores of these variable stars, and the stars' outer atmospheres end up becoming unstable [3]. First, the stars' atmospheres expand, but then the gas becomes more transparent and cools, causing the gas layers to contract. However, the gas as it contracts becomes more opaque and warms up, and the stars' outer atmospheres expand again. This cycle, which takes several months for a typical Mira-type variable star, repeats itself for a very long time.
Chi Cygni itself was discovered to be a Mira-type variable star in the seventeenth century by the German astronomer Gottfried Kirch [4, 5, 6]. He was also known for several other astronomical discoveries, some of which were actually by his wife Maria [7], but I think the discovery of variability in Chi Cygni was by Gottfried himself partly because he did it a few years before he got married. One night in July 1686, Gottfried was looking at an area around the nova CK Vulpeculae and noticed that Chi Cygni was missing. However, when he looked again in October 1686, Chi Cygni had reappeared. He then started monitoring the star and determined that it brightened and dimmed over a very long period of time that he measured to be 405 days [4]. This cycle has been more accurately measured to be 408 days by modern astronomers [6], although the length of this cycle could vary by a few days from cycle to cycle [5]. Impressively, when Gottfried Kirch died in 1710, his wife Maria and his son Christfried continued to watch the star brighten and fade until 1738 [4]. The star has been monitored by other astronomers ever since.
One of the most impressive things about Chi Cygni is that it is one of the brightest Mira variable star in the sky as seen from Earth [6]. (OK, that doesn't quite make sense for something to be both the brightest thing in the sky and variable in brightness at the same time, so let's say that Chi Cygni is one of the Mira variable stars that gets the brightest in the sky as seen from Earth. Hopefully, that makes sense.) Even more impressively, though, its variations in brightness are very extreme compared to other Mira variable stars as well as compared to other types of variable stars in general [6].
At its brightest, Chi Cygni can reach a magnitude of about 3.3, which is around 2 magnitudes brighter than the faintest stars that can be seen without a telescope, but at its faintest, Chi Cygni can drop to about a magnitude of 14.2, which is about 8 magnitudes fainter than what can be seen with the naked eye [6]. (Also, just to remind people, larger magnitudes in astronomy correspond to fainter things, and smaller magnitudes correspond to brighter things.) In physical terms, these variations in magnitude translate into variations of a factor of over 20000 in how much light is emitted by the star. So, to get an idea of how extreme this is, imagine the difference in brightness between 1 LED light and 20000 LED lights, and you get an idea of how much Chi Cygni varies in brightness. (Also, please write in if you actually try comparing the brightness of 1 LED light to 20000 LED lights at home.) Having said all of this, the variations in brightness are themselves variable, so Chi Cygni will not necessarily always get bright enough to see without a telescope or might not get as faint as magnitude 14 [4]. If you're getting confused by this discussion about a variable star where both the period of the variations in brightness can vary and where the variations in brightness can vary, just remember that Chi Cygni varies in brightness an extreme amount and that you can sometimes see it in the sky without a telescope.
Because Chi Cygni at its faintest can still be seen with a halfway-decent amateur telescope, the star is a very popular target for very patient amateur astronomers who are interested in variable stars. In fact, the American Association of Variable Star Observers, among other organizations, coordinates efforts by amateur astronomers to collect data on Chi Cygni and other Mira-type variable stars and maintains databases with information about how these stars vary in brightness, and I would encourage you to visit their website at www.aavso.org for more information about variable stars.
Even though Chi Cygni is a popular amateur astronomy object, it still attracts a lot of attention from professional astronomers as well. In recent research, for example, astronomers using advanced imaging techniques have been able to determine that the star varies in size from about 330 to 480 times the diameter of the Sun [8]. Astronomers interested in how stars produce heavy elements through fusion and how they form molecules in their outer atmospheres from these elements have observed Chi Cygni at submillimeter wavelengths (which, as it sounds, is a form of electromagnetic radiation with wavelengths slightly less than 1 millimeter in size), and they have found water, carbon monoxide, and hydrogen cyanide among other molecules in the atmosphere of this star [9, 10]. Also, some stellar astronomers had wondered why they had difficulty measuring magnetic fields around Mira-type variable stars, so they decided to focus of Chi Cygni because it's so bright, and they eventually were able to measure the magnetic field of the star, although it was really weak. This turned out to be the first time anyone had been able to measure the magnetic field of any Mira-type variable star [11].