Short answers, pulled from the story.
Edmond Halley first recognised the comet as periodic in his 1705 Synopsis of the Astronomy of Comets, after comparing observations from 1531, 1607, and 1682 and concluding all three were the same object. The prediction was confirmed when the comet returned on the 25th of December 1758, spotted by German farmer and amateur astronomer Johann Georg Palitzsch.
In 837 AD, Halley's Comet passed approximately 0.03 astronomical units from Earth, by far its closest recorded approach, equivalent to roughly 5 million kilometres. Its tail may have stretched 60 degrees across the sky, and it was recorded by astronomers in China, Japan, Germany, the Byzantine Empire, and the Middle East.
The Giotto and Vega probes found that Halley's nucleus is roughly 15 kilometres long and 8 kilometres wide, with an albedo of only 0.04 - about as reflective as coal. Surface temperatures measured by Vega 1 ran between 300 and 400 Kelvin, and only about 10% of the surface was active, with the rest coated in dark non-volatile dust, making the comet more a "snowy dirtball" than the predicted "dirty snowball".
Halley's Comet is associated with two annual meteor showers: the Eta Aquariids, which occur in early May, and the Orionids, which occur in late October. Both arise because the comet's orbit crosses Earth's in two places.
The next perihelion of Halley's Comet is predicted for the 28th of July 2061. It is expected to reach an apparent magnitude of -0.3, significantly brighter than its 1986 apparition, and will be on the same side of the Sun as Earth, making for far better viewing conditions.
The 1066 apparition coincided with the Norman Conquest of England: Harold II died at the Battle of Hastings and William the Conqueror took the throne. Witnesses described the comet as appearing four times the size of Venus, and it was depicted on the Bayeux Tapestry. The monk Eilmer of Malmesbury, who may have seen it in both 989 and 1066, reportedly addressed it directly as "a source of tears to many mothers."