Mira, Omicron Ceti, is an oscillating variable star. The first non-supernova variable star discovered, excepting the odd Eta Carinae, it is the brightest periodic variable in the sky that is not visible during part of its cycle.
Mira was discovered (or at least noted as special for the first time) after a series of observations by the astronomer David Fabricius beginning on August 3, 1596. Observing the planet Mercury, he needed a reference star for comparing positions and picked a previously unremarked third-magnitude star nearby. By August 21, however, it had increased in brightness by one magnitude, then by October had faded from view. Fabricius assumed it was a nova -- interesting enough -- but then saw it again on February 16, 1609.
Eventually, Johann Holwarda determined a period of the star's reappearances, eleven months; Johannes Hevelius was observing it at the same time and named it "Mira" (meaning "Wonderful") in 1662's Historiola Mirae Stellae, for it acted like no other known star in the sky. Ismail Bouillaud then nailed the period down to 333 days, less than one day off of the modern value of 332 days (and perfectly forgivable, as Mira is known to vary slightly in period, and may even be slowly changing over time).
From this discovery, Mira became the "type" star of the "Long period variables". It -- and the other 6000 or so known stars of this class -- are all red giants whose surfaces oscillate in such a way as to increase and decrease in brightness over periods ranging from about 80 days to more than 1000. In the particular case of Mira, its increases in brightness take it up to about magnitude 3.5 on average, which is a fairly noticeable star. Individual cycles vary too; well-attested maxima go as high as magnitude 2.0 in brightness and as low as 4.9, a range almost 15 times in brightness, and there are historical suggestions that the real spread may be three times this or more. Minima range much less, and have historically been between 8.6 and 10.1, a factor of four times in luminosity. The total swing in brightness from absolute maximum to absolute minimum (two events which did not occur on the same cycle) is 1700 times. The shape of its light curve is of an increase over about 100 days, and a return twice as long.
Mira is also a double star. The companion star was resolved by the Hubble space telescope in 1995, when it was 70 astronomical units from the primary; results were announced in 1997. The companion is a variable star itself, CZ Ceti, and the best interpretation of this and other data is that it is a white dwarf surrounded by an accretion disk of material drawn off of the primary; the HST ultraviolet images show a spiral of gas rising off Mira in the direction of CZ Ceti. The companion's orbital period around Mira is approximately 400 years.
There is considerable speculation as to whether or not Mira had been observed prior to Fabricius. Certainly Algol's history (known for certain as a variable only in 1667, but with legends and such stretching back to antiquity showing that it had been observed with suspicion for millennia) suggested that Mira might have been known too. Karl Manitius, a translator of Hipparchus' Commentary on Aratus, has suggested that certain lines from that second century BC text may be about Mira. The other usual pre-telescopic Western suspects -- the catalogs of Ptolemy, al-Sufi, Ulugh Beg, and Tycho Brahe -- turn up no mentions, even as a regular star. There are three observations from Chinese and Korean archives, in 1596, 1070, and the same year when Hipparchus would have made his observation (134 BC) that are suggestive, but the Chinese practice of pinning down observations no more precisely than within a given Chinese constellation makes it difficult to be sure.