Short answers, pulled from the story.
The original Death Star is a spherical battle station 160 kilometers in diameter, capable of destroying a planet with a single shot from its superlaser. Sources disagree on the second Death Star's size, with figures ranging from 160 to 900 kilometers in radius, though the most recent figure, set in 2017 by Ryder Windham, gives it a 200 kilometer radius.
Luke Skywalker destroyed the first Death Star by firing his X-wing's torpedoes into a ray-shielded exhaust vent that led to the station's reactor core. The Rebel Alliance located that weakness using schematics Princess Leia had carried to them, and the resulting explosion destroyed the station before it could fire on the Rebel base on the moon Yavin 4.
The Galactic Empire built the Death Star, with engineers led by Orson Krennic and weapons designer Galen Erso developing the project on the world Eadu. Krennic forcibly recruited Erso, the father of Rogue One protagonist Jyn Erso, to complete a design based on fusing kyber crystal shards into a planet-destroying beam.
A Rebel ground team led by Han Solo, aided by the native Ewoks of the forest moon Endor, disabled the second Death Star's shield generator. Rebel pilots Wedge Antilles and Lando Calrissian then flew the Millennium Falcon into the station and fired on its reactor, destroying it in the resulting explosion.
Starkiller Base is a superweapon introduced in The Force Awakens, built by the First Order out of an existing planet called Ilum rather than assembled in space. It is significantly more powerful than either Death Star, drawing its firepower from a nearby star and destroying multiple planets at once, including the five worlds of the Hosnian Prime system.
AT&T Corporation's logo, designed by Saul Bass and introduced in 1982, is informally nicknamed the Death Star because of its visual resemblance to the fictional station. The nickname has also been applied to real-world subjects including Dallas's Cowboys Stadium and a 2012 petition on the White House website asking the government to build a real one.