Short answers, pulled from the story.
Caligula, whose full name was Gaius Julius Caesar Augustus Germanicus, was the third Roman emperor. He ruled from AD 37 until his assassination on the 24th of January AD 41, a reign of roughly four years.
Caligula was a nickname given to him as a young child by the soldiers of his father Germanicus. He accompanied his father on campaigns in Germania at age two or three, and his mother dressed him in a miniature soldier's outfit including army boots called caligae. The soldiers called him Caligula, meaning 'little boot'. He later grew to dislike the name.
Caligula began construction of two aqueducts, the Aqua Claudia and the Anio Novus, which Pliny the Elder considered engineering marvels. He also completed the temple of Augustus, rebuilt the theatre of Pompey, built a racetrack known as the Circus of Gaius and Nero, and transported the Egyptian obelisk now known as the Vatican obelisk to Rome by sea on a purpose-built ship.
Caligula introduced taxes on taverns, artisans, slaves and the hiring of slaves, food sold in the city, litigation across the Empire, weddings, porters' wages, and a tax on prostitutes. He also removed previous tax exemptions from citizens of provincial Italy. Taxes roughly doubled overall during his reign, and the tax on prostitution continued up to the reign of Severus Alexander.
Suetonius and Dio record the story that Caligula planned to make his favourite racehorse Incitatus, whose name meant 'Swift', a consul and later a priest of his own cult. Most modern historians read this as a deliberate satirical insult aimed at the Senate rather than a genuine intention. Historian Winterling sees it as a mockery of the consulship itself as the highest aristocratic ambition.
In early AD 41, Caligula was killed in a conspiracy involving officers of the Praetorian Guard, senators, and courtiers. At least some of the plotters aimed to restore the Roman Republic, but the Praetorian Guard appears to have spontaneously chosen Caligula's uncle Claudius as the next emperor, ending any chance of a republican restoration.